{
"query": {
"display": "$$\\int\\:\\sech\\left(\\frac{1}{4}\\right)x\\tanh\\left(\\frac{1}{4}\\right)xdx$$",
"symbolab_question": "BIG_OPERATOR#\\int \\sech(\\frac{1}{4})x\\tanh(\\frac{1}{4})xdx"
},
"solution": {
"level": "PERFORMED",
"subject": "Calculus",
"topic": "Integrals",
"subTopic": "Indefinite Integrals",
"default": "\\frac{2e^{\\frac{3}{4}}-2\\sqrt[4]{e}}{e+2\\sqrt[4]{e}\\sqrt[4]{e}+1}\\cdot \\frac{x^{3}}{3}+C",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "$$\\int\\:\\sech\\left(\\frac{1}{4}\\right)x\\tanh\\left(\\frac{1}{4}\\right)xdx=\\frac{2e^{\\frac{3}{4}}-2\\sqrt[4]{e}}{e+2\\sqrt[4]{e}\\sqrt[4]{e}+1}\\cdot\\:\\frac{x^{3}}{3}+C$$",
"input": "\\int\\:\\sech\\left(\\frac{1}{4}\\right)x\\tanh\\left(\\frac{1}{4}\\right)xdx",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$",
"result": "=\\sech\\left(\\frac{1}{4}\\right)\\tanh\\left(\\frac{1}{4}\\right)\\cdot\\:\\int\\:xxdx"
},
{
"type": "interim",
"title": "$$\\sech\\left(\\frac{1}{4}\\right)\\tanh\\left(\\frac{1}{4}\\right)=\\frac{2e^{\\frac{3}{4}}-2\\sqrt[4]{e}}{e+2e^{\\frac{1}{4}}\\sqrt[4]{e}+1}$$",
"input": "\\sech\\left(\\frac{1}{4}\\right)\\tanh\\left(\\frac{1}{4}\\right)",
"steps": [
{
"type": "interim",
"title": "$$\\sech\\left(\\frac{1}{4}\\right)=\\frac{2\\sqrt[4]{e}}{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}$$",
"input": "\\sech\\left(\\frac{1}{4}\\right)",
"result": "=\\frac{2\\sqrt[4]{e}}{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}\\tanh\\left(\\frac{1}{4}\\right)",
"steps": [
{
"type": "step",
"primary": "Use the Hyperbolic identity: $$\\sech\\left(x\\right)=\\frac{2}{e^{x}+e^{-x}}$$",
"result": "=\\frac{2}{e^{\\frac{1}{4}}+e^{-\\frac{1}{4}}}"
},
{
"type": "interim",
"title": "$$\\frac{2}{e^{\\frac{1}{4}}+e^{-\\frac{1}{4}}}=\\frac{2\\sqrt[4]{e}}{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}$$",
"input": "\\frac{2}{e^{\\frac{1}{4}}+e^{-\\frac{1}{4}}}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$",
"result": "=\\frac{2}{e^{\\frac{1}{4}}+\\frac{1}{\\sqrt[4]{e}}}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "interim",
"title": "Join $$e^{\\frac{1}{4}}+\\frac{1}{\\sqrt[4]{e}}:{\\quad}\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}{\\sqrt[4]{e}}$$",
"input": "e^{\\frac{1}{4}}+\\frac{1}{\\sqrt[4]{e}}",
"result": "=\\frac{2}{\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}{\\sqrt[4]{e}}}",
"steps": [
{
"type": "step",
"primary": "Convert element to fraction: $$e^{\\frac{1}{4}}=\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}}{\\sqrt[4]{e}}$$",
"result": "=\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}}{\\sqrt[4]{e}}+\\frac{1}{\\sqrt[4]{e}}"
},
{
"type": "step",
"primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$",
"result": "=\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}{\\sqrt[4]{e}}"
}
],
"meta": {
"interimType": "Algebraic Manipulation Join Concise Title 1Eq"
}
},
{
"type": "step",
"primary": "Apply the fraction rule: $$\\frac{a}{\\frac{b}{c}}=\\frac{a\\cdot\\:c}{b}$$",
"result": "=\\frac{2\\sqrt[4]{e}}{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s75gOnbYE5qv86GPQZJohQUK5+EerJMzTX31f9y2ha+rgyzizhCCAZHgGPHLZz0Tn9LTrWWMFI8l4Q07DZ5+hJa0hqcfXty6JYd1pWIgQ3ZjiTEfPyE5xYl8qdYjdUIGOTd1WMtrHLa5W8WuK2lk/qiYC10zhmuDOvJ5vCD8uvt8X/P/+v51eTuH2/F4MIu5mqhjVnk05ZDMy7kNtKK3LJfCjnEzKjKCXaEepC4tBDFN283OCDj+JDig38UyGcm4VMmLz+W2iJCI8WrzJTYd442WN6S1jJsg4ygZpQohy68eihAQ5ImRWNnI4QzVRUuoBPnTYklRHwD3igJbr9RjBhPw=="
}
},
{
"type": "step",
"result": "=\\frac{2\\sqrt[4]{e}}{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}"
}
],
"meta": {
"solvingClass": "Trig Evaluate",
"interimType": "Trig Evaluate"
}
},
{
"type": "interim",
"title": "$$\\tanh\\left(\\frac{1}{4}\\right)=\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}-1}{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}$$",
"input": "\\tanh\\left(\\frac{1}{4}\\right)",
"result": "=\\frac{2\\sqrt[4]{e}}{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}\\cdot\\:\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}-1}{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}",
"steps": [
{
"type": "step",
"primary": "Use the Hyperbolic identity: $$\\tanh\\left(x\\right)=\\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$$",
"result": "=\\frac{e^{\\frac{1}{4}}-e^{-\\frac{1}{4}}}{e^{\\frac{1}{4}}+e^{-\\frac{1}{4}}}"
},
{
"type": "interim",
"title": "$$\\frac{e^{\\frac{1}{4}}-e^{-\\frac{1}{4}}}{e^{\\frac{1}{4}}+e^{-\\frac{1}{4}}}=\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}-1}{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}$$",
"input": "\\frac{e^{\\frac{1}{4}}-e^{-\\frac{1}{4}}}{e^{\\frac{1}{4}}+e^{-\\frac{1}{4}}}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$",
"result": "=\\frac{e^{\\frac{1}{4}}-e^{-\\frac{1}{4}}}{e^{\\frac{1}{4}}+\\frac{1}{\\sqrt[4]{e}}}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Apply exponent rule: $$a^{-b}=\\frac{1}{a^b}$$",
"result": "=\\frac{e^{\\frac{1}{4}}-\\frac{1}{\\sqrt[4]{e}}}{e^{\\frac{1}{4}}+\\frac{1}{\\sqrt[4]{e}}}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "interim",
"title": "Join $$e^{\\frac{1}{4}}+\\frac{1}{\\sqrt[4]{e}}:{\\quad}\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}{\\sqrt[4]{e}}$$",
"input": "e^{\\frac{1}{4}}+\\frac{1}{\\sqrt[4]{e}}",
"result": "=\\frac{e^{\\frac{1}{4}}-\\frac{1}{\\sqrt[4]{e}}}{\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}{\\sqrt[4]{e}}}",
"steps": [
{
"type": "step",
"primary": "Convert element to fraction: $$e^{\\frac{1}{4}}=\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}}{\\sqrt[4]{e}}$$",
"result": "=\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}}{\\sqrt[4]{e}}+\\frac{1}{\\sqrt[4]{e}}"
},
{
"type": "step",
"primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$",
"result": "=\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}{\\sqrt[4]{e}}"
}
],
"meta": {
"interimType": "Algebraic Manipulation Join Concise Title 1Eq"
}
},
{
"type": "interim",
"title": "Join $$e^{\\frac{1}{4}}-\\frac{1}{\\sqrt[4]{e}}:{\\quad}\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}-1}{\\sqrt[4]{e}}$$",
"input": "e^{\\frac{1}{4}}-\\frac{1}{\\sqrt[4]{e}}",
"result": "=\\frac{\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}-1}{\\sqrt[4]{e}}}{\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}{\\sqrt[4]{e}}}",
"steps": [
{
"type": "step",
"primary": "Convert element to fraction: $$e^{\\frac{1}{4}}=\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}}{\\sqrt[4]{e}}$$",
"result": "=\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}}{\\sqrt[4]{e}}-\\frac{1}{\\sqrt[4]{e}}"
},
{
"type": "step",
"primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$",
"result": "=\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}-1}{\\sqrt[4]{e}}"
}
],
"meta": {
"interimType": "Algebraic Manipulation Join Concise Title 1Eq"
}
},
{
"type": "step",
"primary": "Divide fractions: $$\\frac{\\frac{a}{b}}{\\frac{c}{d}}=\\frac{a\\cdot\\:d}{b\\cdot\\:c}$$",
"result": "=\\frac{\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}-1\\right)\\sqrt[4]{e}}{\\sqrt[4]{e}\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}+1\\right)}"
},
{
"type": "step",
"primary": "Cancel the common factor: $$\\sqrt[4]{e}$$",
"result": "=\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}-1}{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "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"
}
},
{
"type": "step",
"result": "=\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}-1}{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}"
}
],
"meta": {
"solvingClass": "Trig Evaluate",
"interimType": "Trig Evaluate"
}
},
{
"type": "interim",
"title": "Simplify",
"input": "\\frac{2\\sqrt[4]{e}}{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}\\cdot\\:\\frac{e^{\\frac{1}{4}}\\sqrt[4]{e}-1}{e^{\\frac{1}{4}}\\sqrt[4]{e}+1}",
"steps": [
{
"type": "step",
"primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$",
"result": "=\\frac{2\\sqrt[4]{e}\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}-1\\right)}{\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}+1\\right)\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}+1\\right)}"
},
{
"type": "interim",
"title": "$$\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}+1\\right)\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}+1\\right)=\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}+1\\right)^{2}$$",
"input": "\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}+1\\right)\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}+1\\right)",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}+1\\right)\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}+1\\right)=\\:\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}+1\\right)^{1+1}$$"
],
"result": "=\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}+1\\right)^{1+1}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add the numbers: $$1+1=2$$",
"result": "=\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}+1\\right)^{2}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7q6XXxzXXkiNRIGBYcem6IuGZhwnS+mDfYNp23/Fc4NrqqSvqHzki8bKgXG+lHktV4ZmHCdL6YN9g2nbf8Vzg2jxqATCUZ7VcdLupGuYinH0AasAFQu4+COAlZGg3sK3EchdOQz/XuPw/noc2GBUIc3mklUck2FHCDd/dXeh+f1NeTZMykILbMC5S4vTIC/oKZ86Ysj73JTHkWzFpzNhE+WsdXh/GPn9ubTv2pBRrh4KTHnFXRHsB4wq4WLs5iqb5ax1eH8Y+f25tO/akFGuHguFWR48kWIuaJNLwpE/9N+ahAQ5ImRWNnI4QzVRUuoBPg7qVOWl8Cmium0CRK+0g+A=="
}
},
{
"type": "step",
"result": "=\\frac{2\\sqrt[4]{e}\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}-1\\right)}{\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}+1\\right)^{2}}"
},
{
"type": "interim",
"title": "$$\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}+1\\right)^{2}=e+2e^{\\frac{1}{4}}\\sqrt[4]{e}+1$$",
"input": "\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}+1\\right)^{2}",
"steps": [
{
"type": "step",
"primary": "Apply Perfect Square Formula: $$\\left(a+b\\right)^{2}=a^{2}+2ab+b^{2}$$",
"secondary": [
"$$a=e^{\\frac{1}{4}}\\sqrt[4]{e},\\:\\:b=1$$"
],
"meta": {
"practiceLink": "/practice/expansion-practice#area=main&subtopic=Perfect%20Square",
"practiceTopic": "Expand Perfect Square"
}
},
{
"type": "step",
"result": "=\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}\\right)^{2}+2e^{\\frac{1}{4}}\\sqrt[4]{e}\\cdot\\:1+1^{2}"
},
{
"type": "interim",
"title": "Simplify $$\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}\\right)^{2}+2e^{\\frac{1}{4}}\\sqrt[4]{e}\\cdot\\:1+1^{2}:{\\quad}e+2e^{\\frac{1}{4}}\\sqrt[4]{e}+1$$",
"input": "\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}\\right)^{2}+2e^{\\frac{1}{4}}\\sqrt[4]{e}\\cdot\\:1+1^{2}",
"result": "=e+2e^{\\frac{1}{4}}\\sqrt[4]{e}+1",
"steps": [
{
"type": "step",
"primary": "Apply rule $$1^{a}=1$$",
"secondary": [
"$$1^{2}=1$$"
],
"result": "=\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}\\right)^{2}+2\\cdot\\:1\\cdot\\:e^{\\frac{1}{4}}\\sqrt[4]{e}+1"
},
{
"type": "interim",
"title": "$$\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}\\right)^{2}=e$$",
"input": "\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}\\right)^{2}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(a\\cdot\\:b\\right)^{n}=a^{n}b^{n}$$",
"result": "=\\left(\\sqrt[4]{e}\\right)^{2}\\left(e^{\\frac{1}{4}}\\right)^{2}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "interim",
"title": "$$\\left(e^{\\frac{1}{4}}\\right)^{2}:{\\quad}\\sqrt{e}$$",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$",
"result": "=e^{\\frac{1}{4}\\cdot\\:2}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "interim",
"title": "$$\\frac{1}{4}\\cdot\\:2=\\frac{1}{2}$$",
"input": "\\frac{1}{4}\\cdot\\:2",
"result": "=e^{\\frac{1}{2}}",
"steps": [
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{1\\cdot\\:2}{4}"
},
{
"type": "step",
"primary": "Multiply the numbers: $$1\\cdot\\:2=2$$",
"result": "=\\frac{2}{4}"
},
{
"type": "step",
"primary": "Cancel the common factor: $$2$$",
"result": "=\\frac{1}{2}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/+LVvf+dQpmEy47zPJ1c6e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBZsqxqhl2a6oRKVJk8034tWRLd2VwIqlBNByF6663syQBIKwn7K8QnbTaJHiTcagGlM87K6KapH6E0P0iYBWsnbCI2sSeA74029n2yo277ZU="
}
},
{
"type": "step",
"primary": "$$a^{\\frac{1}{2}}=\\sqrt{a}$$",
"result": "=\\sqrt{e}"
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "step",
"result": "=\\sqrt{e}\\left(\\sqrt[4]{e}\\right)^{2}"
},
{
"type": "interim",
"title": "$$\\left(\\sqrt[4]{e}\\right)^{2}:{\\quad}\\sqrt{e}$$",
"steps": [
{
"type": "step",
"primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$",
"result": "=\\left(e^{\\frac{1}{4}}\\right)^{2}",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
},
{
"type": "step",
"primary": "Apply exponent rule: $$\\left(a^{b}\\right)^{c}=a^{bc}$$",
"result": "=e^{\\frac{1}{4}\\cdot\\:2}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "interim",
"title": "$$\\frac{1}{4}\\cdot\\:2=\\frac{1}{2}$$",
"input": "\\frac{1}{4}\\cdot\\:2",
"result": "=e^{\\frac{1}{2}}",
"steps": [
{
"type": "step",
"primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$",
"result": "=\\frac{1\\cdot\\:2}{4}"
},
{
"type": "step",
"primary": "Multiply the numbers: $$1\\cdot\\:2=2$$",
"result": "=\\frac{2}{4}"
},
{
"type": "step",
"primary": "Cancel the common factor: $$2$$",
"result": "=\\frac{1}{2}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7/+LVvf+dQpmEy47zPJ1c6e30Fwl9QKPJxyO/TFRCb5Grju+5Z51e/ZZSD3gRHwjBZsqxqhl2a6oRKVJk8034tWRLd2VwIqlBNByF6663syQBIKwn7K8QnbTaJHiTcagGlM87K6KapH6E0P0iYBWsnbCI2sSeA74029n2yo277ZU="
}
},
{
"type": "step",
"primary": "$$a^{\\frac{1}{2}}=\\sqrt{a}$$",
"result": "=\\sqrt{e}"
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "step",
"result": "=\\sqrt{e}\\sqrt{e}"
},
{
"type": "step",
"primary": "Apply radical rule: $$\\sqrt{a}\\sqrt{a}=a$$",
"secondary": [
"$$\\sqrt{e}\\sqrt{e}=e$$"
],
"result": "=e",
"meta": {
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7q6XXxzXXkiNRIGBYcem6IrbcuEoH7zEECWmxM1RxO6KOSKXZYhtROCuBLQxizfcrzMFYmi1F5Hg/ibpEToVnY/XC6/K3hrpdFZ3cV8OPO6DCefEKAVbuWDr+fw9aKACptty4SgfvMQQJabEzVHE7omFNSt++nTMOe3QiXFqh/tA="
}
},
{
"type": "interim",
"title": "$$2e^{\\frac{1}{4}}\\sqrt[4]{e}\\cdot\\:1=2e^{\\frac{1}{4}}\\sqrt[4]{e}$$",
"input": "2e^{\\frac{1}{4}}\\sqrt[4]{e}\\cdot\\:1",
"steps": [
{
"type": "step",
"primary": "Multiply the numbers: $$2\\cdot\\:1=2$$",
"result": "=2e^{\\frac{1}{4}}\\sqrt[4]{e}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7zzFfjCXTLjuWM/bluMl7X6cHR381I6qeQo0tmJiS3jCkpzcxcRcRYMsMU8GLLSOfCUCWbkwGOY7PqKo3U/JLJUoG1v2HGypTCaqX2ZB+NI2hAQ5ImRWNnI4QzVRUuoBP/z//r+dXk7h9vxeDCLuZqtU27sBC776kPWEg6IgHY3EqYv4Gz9eGoM+AGlGZeUui8R2XvzMGJfvnhGYBwRK8VmEoR7/PvD9fqeq9+d0+XOYKlT/F2wT4M19lmszyiIMB"
}
},
{
"type": "step",
"result": "=e+2e^{\\frac{1}{4}}\\sqrt[4]{e}+1"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7q6XXxzXXkiNRIGBYcem6IuGZhwnS+mDfYNp23/Fc4No81/vVEBMIJkzLClFtiofOq47vuWedXv2WUg94ER8IwV4SySwlh4HfQXkpzmS7Buh3pf8+1DnYez5hv6UgWxp0FByzK4UX45YMEkYvkRgAG41QLhtIiAJe6pN76sLZPh2hAQ5ImRWNnI4QzVRUuoBPiG9IGQn2HF7q1TSn64zEWjXn7cwMLH/UdiwnfRKbQ5OEA+LkkxB0dsBNartfhCEp"
}
},
{
"type": "step",
"result": "=\\frac{2\\sqrt[4]{e}\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}-1\\right)}{e+2e^{\\frac{1}{4}}\\sqrt[4]{e}+1}"
},
{
"type": "interim",
"title": "Expand $$2\\sqrt[4]{e}\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}-1\\right):{\\quad}2e^{\\frac{3}{4}}-2\\sqrt[4]{e}$$",
"input": "2\\sqrt[4]{e}\\left(e^{\\frac{1}{4}}\\sqrt[4]{e}-1\\right)",
"result": "=\\frac{2e^{\\frac{3}{4}}-2\\sqrt[4]{e}}{e+2e^{\\frac{1}{4}}\\sqrt[4]{e}+1}",
"steps": [
{
"type": "step",
"primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$",
"secondary": [
"$$a=2\\sqrt[4]{e},\\:b=e^{\\frac{1}{4}}\\sqrt[4]{e},\\:c=1$$"
],
"result": "=2\\sqrt[4]{e}e^{\\frac{1}{4}}\\sqrt[4]{e}-2\\sqrt[4]{e}\\cdot\\:1",
"meta": {
"practiceLink": "/practice/expansion-practice",
"practiceTopic": "Expand Rules"
}
},
{
"type": "step",
"result": "=2\\sqrt[4]{e}e^{\\frac{1}{4}}\\sqrt[4]{e}-2\\cdot\\:1\\cdot\\:\\sqrt[4]{e}"
},
{
"type": "interim",
"title": "Simplify $$2\\sqrt[4]{e}e^{\\frac{1}{4}}\\sqrt[4]{e}-2\\cdot\\:1\\cdot\\:\\sqrt[4]{e}:{\\quad}2e^{\\frac{3}{4}}-2\\sqrt[4]{e}$$",
"input": "2\\sqrt[4]{e}e^{\\frac{1}{4}}\\sqrt[4]{e}-2\\cdot\\:1\\cdot\\:\\sqrt[4]{e}",
"result": "=2e^{\\frac{3}{4}}-2\\sqrt[4]{e}",
"steps": [
{
"type": "interim",
"title": "$$2\\sqrt[4]{e}e^{\\frac{1}{4}}\\sqrt[4]{e}=2e^{\\frac{3}{4}}$$",
"input": "2\\sqrt[4]{e}e^{\\frac{1}{4}}\\sqrt[4]{e}",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$\\sqrt[4]{e}\\sqrt[4]{e}=\\:e^{\\frac{1}{4}}e^{\\frac{1}{4}}=\\:e^{\\frac{1}{4}+\\frac{1}{4}}$$"
],
"result": "=2e^{\\frac{1}{4}}e^{\\frac{1}{4}+\\frac{1}{4}}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$e^{\\frac{1}{4}}e^{\\frac{1}{4}+\\frac{1}{4}}=\\:e^{\\frac{1}{4}+\\frac{1}{4}+\\frac{1}{4}}$$"
],
"result": "=2e^{\\frac{1}{4}+\\frac{1}{4}+\\frac{1}{4}}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "interim",
"title": "$$e^{\\frac{1}{4}+\\frac{1}{4}+\\frac{1}{4}}=e^{\\frac{3}{4}}$$",
"input": "e^{\\frac{1}{4}+\\frac{1}{4}+\\frac{1}{4}}",
"steps": [
{
"type": "interim",
"title": "Combine the fractions $$\\frac{1}{4}+\\frac{1}{4}+\\frac{1}{4}:{\\quad}\\frac{3}{4}$$",
"result": "=e^{\\frac{3}{4}}",
"steps": [
{
"type": "step",
"primary": "Apply rule $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$",
"result": "=\\frac{1+1+1}{4}"
},
{
"type": "step",
"primary": "Add the numbers: $$1+1+1=3$$",
"result": "=\\frac{3}{4}"
}
],
"meta": {
"interimType": "LCD Top Title 1Eq"
}
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7vOajRfB2Rx64LpD7WyzCEvUK9vPUISoLzYtvMX6yAO+6XHwKS+dS19k1ubbWfbGvICf2WQN9mSJxQaQ7cQX4iqTOspmQmjCdN72k06RUjruH0chMdd1TPT8fWow9fcUsIF3DEFKpqblBA5/jiPK5umQ/7xqawY8LPmju3CRbCmMFcME871kcyQapIde8jtKKs1Ul+KJ3FNayUB5y7y7dfLhrMEJk4Ztoc1O5iFg5r34="
}
},
{
"type": "step",
"result": "=2e^{\\frac{3}{4}}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7u1LOvDzW49vtlghZd+ptewESHE1ioRhKe0p9vXaA0D/lFZAr31hAZhNLrn7/0ZgL3XeO2tIUPH5Q2xrCOU6NXXFu/nSMYeaCe7F3KogWofyH0chMdd1TPT8fWow9fcUsI62gMlfwhukWv8aNVqdLwZD+4Xqz05Mt4Iso9+yUYKJ3pf8+1DnYez5hv6UgWxp03oRdFMBrtg/zcUODX0+JRbhrMEJk4Ztoc1O5iFg5r34="
}
},
{
"type": "interim",
"title": "$$2\\cdot\\:1\\cdot\\:\\sqrt[4]{e}=2\\sqrt[4]{e}$$",
"input": "2\\cdot\\:1\\cdot\\:\\sqrt[4]{e}",
"steps": [
{
"type": "step",
"primary": "Multiply the numbers: $$2\\cdot\\:1=2$$",
"result": "=2\\sqrt[4]{e}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ht1jgkupea8+jUgT+M/VFJ9lTPOc97ut3y3uTmo0sWVV00rpv8+ZC6TM10tVCSHsd1m+ZdxyQp/vq78KeLkuh4UPO9tqR80GFUFB3LyB+xQx829OfivBHYoiC5AiKMnbndzWMyTuzLjJbNVwhWQrYyb3PHL/yOtPR4yYUPxpdT3wYHkfmHHTze3JXk3LjiJp"
}
},
{
"type": "step",
"result": "=2e^{\\frac{3}{4}}-2\\sqrt[4]{e}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
}
],
"meta": {
"interimType": "Algebraic Manipulation Expand Title 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7R4hBKyz57zC7WQmURiq5K3IXTkM/17j8P56HNhgVCHO2E46P23nqR2jr7UJF3vUmLTrWWMFI8l4Q07DZ5+hJa7iStsfbrvU4nxoioeIHQAikRz01S5VpAffojWUUGz+gtT+ODJs6BHDu1n4f26ACoIEFMST8lDZxn1Yq5HMKVTukTJBN6wcjP2NQ8cX7N8Rcsnj1xKa5fAr9jO19XoiKmKkZe58v12C/erH01diYuxXeuyUNRr5IWmoNcGL2B4EB"
}
}
],
"meta": {
"interimType": "Generic Simplify 0Eq"
}
},
{
"type": "step",
"result": "=\\frac{2e^{\\frac{3}{4}}-2\\sqrt[4]{e}}{e+2e^{\\frac{1}{4}}\\sqrt[4]{e}+1}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Solver"
}
},
{
"type": "step",
"result": "=\\frac{2e^{\\frac{3}{4}}-2\\sqrt[4]{e}}{e+2e^{\\frac{1}{4}}\\sqrt[4]{e}+1}\\cdot\\:\\int\\:xxdx"
},
{
"type": "interim",
"title": "Simplify $$xx:{\\quad}x^{2}$$",
"input": "xx",
"steps": [
{
"type": "step",
"primary": "Apply exponent rule: $$a^b\\cdot\\:a^c=a^{b+c}$$",
"secondary": [
"$$xx=\\:x^{1+1}$$"
],
"result": "=x^{1+1}",
"meta": {
"practiceLink": "/practice/exponent-practice",
"practiceTopic": "Expand FOIL"
}
},
{
"type": "step",
"primary": "Add the numbers: $$1+1=2$$",
"result": "=x^{2}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"result": "=\\frac{2e^{\\frac{3}{4}}-2\\sqrt[4]{e}}{e+2e^{\\frac{1}{4}}\\sqrt[4]{e}+1}\\cdot\\:\\int\\:x^{2}dx"
},
{
"type": "interim",
"title": "Apply the Power Rule",
"input": "\\int\\:x^{2}dx",
"result": "=\\frac{2e^{\\frac{3}{4}}-2\\sqrt[4]{e}}{e+2e^{\\frac{1}{4}}\\sqrt[4]{e}+1}\\cdot\\:\\frac{x^{3}}{3}",
"steps": [
{
"type": "step",
"primary": "Apply the Power Rule: $$\\int{x^{a}}dx=\\frac{x^{a+1}}{a+1},\\:\\quad\\:a\\neq{-1}$$",
"result": "=\\frac{x^{2+1}}{2+1}"
},
{
"type": "interim",
"title": "Simplify $$\\frac{x^{2+1}}{2+1}:{\\quad}\\frac{x^{3}}{3}$$",
"input": "\\frac{x^{2+1}}{2+1}",
"steps": [
{
"type": "step",
"primary": "Add the numbers: $$2+1=3$$",
"result": "=\\frac{x^{3}}{3}"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Algebraic Manipulation Simplify Title 1Eq"
}
},
{
"type": "step",
"result": "=\\frac{x^{3}}{3}"
}
],
"meta": {
"interimType": "Power Rule Top 0Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7+w+ikB2VyJnNfLrQuoxvVyo/JI5bBgpgExN510TA5cyodqSCYnUP+KiNK7E2zlYiE/QYMzREewyYhmRoDar7odVISTIak7VD9OG2tlObqsigQUxJPyUNnGfVirkcwpVOw39JmBCMfU6hqFWM4cbYeuPws1TZ9p9GAZMOucM4Sei"
}
},
{
"type": "step",
"primary": "Apply radical rule: $$a^{\\frac{1}{n}}=\\sqrt[n]{a}$$",
"result": "=\\frac{2e^{\\frac{3}{4}}-2\\sqrt[4]{e}}{e+2\\sqrt[4]{e}\\sqrt[4]{e}+1}\\cdot\\:\\frac{x^{3}}{3}",
"meta": {
"solvingClass": "Solver",
"practiceLink": "/practice/radicals-practice",
"practiceTopic": "Radical Rules"
}
},
{
"type": "step",
"primary": "Add a constant to the solution",
"result": "=\\frac{2e^{\\frac{3}{4}}-2\\sqrt[4]{e}}{e+2\\sqrt[4]{e}\\sqrt[4]{e}+1}\\cdot\\:\\frac{x^{3}}{3}+C",
"meta": {
"title": {
"extension": "If $$\\frac{dF\\left(x\\right)}{dx}=f\\left(x\\right)$$ then $$\\int{f\\left(x\\right)}dx=F\\left(x\\right)+C$$"
}
}
}
],
"meta": {
"solvingClass": "Integrals",
"practiceLink": "/practice/integration-practice#area=main&subtopic=Power%20Rule",
"practiceTopic": "Integral Power Rule"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"plotRequest": "y=\\frac{2e^{\\frac{3}{4}}-2\\sqrt[4]{e}}{e+2\\sqrt[4]{e}\\sqrt[4]{e}+1}\\cdot \\frac{x^{3}}{3}+C"
},
"showViewLarger": true
}
},
"meta": {
"showVerify": true
}
}
Solution
Solution
Solution steps
Take the constant out:
Simplify
Apply the Power Rule
Apply radical rule:
Add a constant to the solution