integral of (4+4x)/(1+x^2)

{ "query": { "display": "$$\\int\\:\\frac{4+4x}{1+x^{2}}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{4+4x}{1+x^{2}}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "4\\arctan(x)+2\\ln\\left|1+x^{2}\\right|+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{4+4x}{1+x^{2}}dx=4\\arctan\\left(x\\right)+2\\ln\\left|1+x^{2}\\right|+C$$", "input": "\\int\\:\\frac{4+4x}{1+x^{2}}dx", "steps": [ { "type": "interim", "title": "Expand $$\\frac{4+4x}{1+x^{2}}:{\\quad}\\frac{4}{1+x^{2}}+\\frac{4x}{1+x^{2}}$$", "input": "\\frac{4+4x}{1+x^{2}}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$", "result": "=\\frac{4}{1+x^{2}}+\\frac{4x}{1+x^{2}}" } ], "meta": { "interimType": "Algebraic Manipulation Expand 1Eq" } }, { "type": "step", "primary": "Apply the Sum Rule: $$\\int{f\\left(x\\right){\\pm}g\\left(x\\right)}dx=\\int{f\\left(x\\right)}dx{\\pm}\\int{g\\left(x\\right)}dx$$", "result": "=\\int\\:\\frac{4}{1+x^{2}}dx+\\int\\:\\frac{4x}{1+x^{2}}dx" }, { "type": "interim", "title": "$$\\int\\:\\frac{4}{1+x^{2}}dx=4\\arctan\\left(x\\right)$$", "input": "\\int\\:\\frac{4}{1+x^{2}}dx", "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": "=4\\cdot\\:\\int\\:\\frac{1}{1+x^{2}}dx" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{1+x^{2}}dx=\\arctan\\left(x\\right)$$", "result": "=4\\arctan\\left(x\\right)" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:\\frac{4x}{1+x^{2}}dx=2\\ln\\left|1+x^{2}\\right|$$", "input": "\\int\\:\\frac{4x}{1+x^{2}}dx", "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": "=4\\cdot\\:\\int\\:\\frac{x}{1+x^{2}}dx" }, { "type": "interim", "title": "Apply u-substitution", "input": "\\int\\:\\frac{x}{1+x^{2}}dx", "steps": [ { "type": "definition", "title": "Integral Substitution definition", "text": "$$\\int\\:f\\left(g\\left(x\\right)\\right)\\cdot\\:g'\\left(x\\right)dx=\\int\\:f\\left(u\\right)du,\\:\\quad\\:u=g\\left(x\\right)$$", "secondary": [ "Substitute: $$u=1+x^{2}$$" ] }, { "type": "interim", "title": "$$\\frac{du}{dx}=2x$$", "input": "\\frac{d}{dx}\\left(1+x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$", "result": "=\\frac{d}{dx}\\left(1\\right)+\\frac{d}{dx}\\left(x^{2}\\right)" }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(1\\right)=0$$", "input": "\\frac{d}{dx}\\left(1\\right)", "steps": [ { "type": "step", "primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmqQX14xoif/Hxcm4iYenIFJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTsKfyXa6Zj1lcQsTYejuhcz" } }, { "type": "interim", "title": "$$\\frac{d}{dx}\\left(x^{2}\\right)=2x$$", "input": "\\frac{d}{dx}\\left(x^{2}\\right)", "steps": [ { "type": "step", "primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$", "result": "=2x^{2-1}", "meta": { "practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule", "practiceTopic": "Power Rule" } }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkmb3s5xAUYje7fZWSRkdb2k3hxk9aCfAWodBRxXgUexcQsmN/cITrVSOMImEqe3fkeCBKuYKgaNJ253gLI69U7cjrVUqImvoUuRtb+2ccCzWsr9JoDNJaP7hueshcYJ6w==" } }, { "type": "step", "result": "=0+2x" }, { "type": "step", "primary": "Simplify", "result": "=2x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives" } }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:du=2xdx$$" }, { "type": "step", "primary": "$$\\quad\\Rightarrow\\:dx=\\frac{1}{2x}du$$" }, { "type": "step", "result": "=\\int\\:\\frac{x}{u}\\cdot\\:\\frac{1}{2x}du" }, { "type": "interim", "title": "Simplify $$\\frac{x}{u}\\cdot\\:\\frac{1}{2x}:{\\quad}\\frac{1}{2u}$$", "input": "\\frac{x}{u}\\cdot\\:\\frac{1}{2x}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$\\frac{a}{b}\\cdot\\frac{c}{d}=\\frac{a\\:\\cdot\\:c}{b\\:\\cdot\\:d}$$", "result": "=\\frac{x\\cdot\\:1}{u\\cdot\\:2x}" }, { "type": "step", "primary": "Cancel the common factor: $$x$$", "result": "=\\frac{1}{u\\cdot\\:2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "=\\int\\:\\frac{1}{2u}du" } ], "meta": { "interimType": "Integral U Substitution 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7l7+Hp9GxvB/5I+4/5L7s7yagDxpghcPiEdNRRng5JS9OYXPdyEhgTWrx7HQOv/Md2NCwimM7dB8C524HvMCB4ymZOc9q9xxqJAg2jt99wha4eveJYYFfQMHyfOzMWdYhwdd2LVu4R2dM/3xRRuc0eEUqTd96MWTKI6Kr2Ib0iQBZegS2gwh8pq/gwNfwaDRbAwNT33I9ftOGdlyhcHnXlA==" } }, { "type": "step", "result": "=4\\cdot\\:\\int\\:\\frac{1}{2u}du" }, { "type": "step", "primary": "Take the constant out: $$\\int{a\\cdot{f\\left(x\\right)}dx}=a\\cdot\\int{f\\left(x\\right)dx}$$", "result": "=4\\cdot\\:\\frac{1}{2}\\cdot\\:\\int\\:\\frac{1}{u}du" }, { "type": "step", "primary": "Use the common integral: $$\\int\\:\\frac{1}{u}du=\\ln\\left(\\left|u\\right|\\right)$$", "result": "=4\\cdot\\:\\frac{1}{2}\\ln\\left|u\\right|" }, { "type": "step", "primary": "Substitute back $$u=1+x^{2}$$", "result": "=4\\cdot\\:\\frac{1}{2}\\ln\\left|1+x^{2}\\right|" }, { "type": "interim", "title": "Simplify $$4\\cdot\\:\\frac{1}{2}\\ln\\left|1+x^{2}\\right|:{\\quad}2\\ln\\left|1+x^{2}\\right|$$", "input": "4\\cdot\\:\\frac{1}{2}\\ln\\left|1+x^{2}\\right|", "result": "=2\\ln\\left|1+x^{2}\\right|", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:4}{2}\\ln\\left|x^{2}+1\\right|" }, { "type": "interim", "title": "$$\\frac{1\\cdot\\:4}{2}=2$$", "input": "\\frac{1\\cdot\\:4}{2}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$1\\cdot\\:4=4$$", "result": "=\\frac{4}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{4}{2}=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CLfSHLR0CGVpdHoWBOzGCgfha7uJwsPTE0M5ka20xmerju+5Z51e/ZZSD3gRHwjBSLdINoPD2MyLmUXT+YnlMgZ7aVYt9I8x8R7JWUC2axAC3g2POzK5Nv1UEqJykGVH" } }, { "type": "step", "result": "=2\\ln\\left|x^{2}+1\\right|" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CBZG1M+6blmUrmpAbZmV80poPwoVN8CD9sMh9OrOY+CJAJeMHQwkTU4NKdEfWLMAAJYpRu9XpYrd8NSAW2DdD1qcvkGoOmj7vUKbPyNbZzGcEA3DuEyFiw5qkhFqoLIOGR/8ixMM14KwAl+dZjPHQnql8XXPq6bNQlMm+36iNhmuWFT1HL/DbhC2ZOlZ+BAa/O4SkJC5+UNVeynAO0/PRu5Ex25Xr9z9ZnoXUrsk0+J2s2pf7n4euhW9r5LxeszL" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=4\\arctan\\left(x\\right)+2\\ln\\left|1+x^{2}\\right|" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=4\\arctan\\left(x\\right)+2\\ln\\left|1+x^{2}\\right|+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=Substitution", "practiceTopic": "Integral Substitution" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=4\\arctan(x)+2\\ln\\left|1+x^{2}\\right|+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }