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

{ "query": { "display": "$$\\int\\:\\frac{1-x^{2}}{1+x^{2}}dx$$", "symbolab_question": "BIG_OPERATOR#\\int \\frac{1-x^{2}}{1+x^{2}}dx" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Integrals", "subTopic": "Indefinite Integrals", "default": "2\\arctan(x)-x+C", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\int\\:\\frac{1-x^{2}}{1+x^{2}}dx=2\\arctan\\left(x\\right)-x+C$$", "input": "\\int\\:\\frac{1-x^{2}}{1+x^{2}}dx", "steps": [ { "type": "interim", "title": "$$\\frac{1-x^{2}}{1+x^{2}}=\\frac{2}{1+x^{2}}-1$$", "input": "\\frac{1-x^{2}}{1+x^{2}}", "steps": [ { "type": "step", "primary": "$$\\frac{1-x^{2}}{1+x^{2}}=\\frac{1-x^{2}+\\left(1+x^{2}\\right)}{1+x^{2}}-1$$", "result": "=\\frac{1-x^{2}+\\left(1+x^{2}\\right)}{1+x^{2}}-1", "meta": { "title": { "extension": "Apply the following algebraic property$$:{\\quad}\\frac{a}{1-a}=\\frac{1}{1-a}-1$$<br/>$$\\frac{a}{1-a}=\\frac{1-1+a}{1-a}=\\frac{1}{1-a}+\\frac{-1+a}{1-a}=\\frac{1}{1-a}+\\frac{-\\left(1-a\\right)}{1-a}=\\frac{1}{1-a}-1$$" } } }, { "type": "interim", "title": "Simplify $$\\frac{1-x^{2}+\\left(1+x^{2}\\right)}{1+x^{2}}-1:{\\quad}\\frac{2}{1+x^{2}}-1$$", "input": "\\frac{1-x^{2}+\\left(1+x^{2}\\right)}{1+x^{2}}-1", "result": "=\\frac{2}{1+x^{2}}-1", "steps": [ { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=\\frac{x^{2}-x^{2}+1+1}{x^{2}+1}-1" }, { "type": "interim", "title": "$$1-x^{2}+1+x^{2}=2$$", "input": "1-x^{2}+1+x^{2}", "steps": [ { "type": "step", "primary": "Group like terms", "result": "=-x^{2}+x^{2}+1+1" }, { "type": "step", "primary": "Add similar elements: $$-x^{2}+x^{2}=0$$", "result": "=1+1" }, { "type": "step", "primary": "Add the numbers: $$1+1=2$$", "result": "=2" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7pZuul1+D7iHgE+YDCoLb8XyRHuGw7+tM5METTDj6vVH6JwS9po8Lx246e07g7nTMjFF+Grhte/2UqFkzsPs4p3iSAKjwbJyigLLck0Jp9aSwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "step", "result": "=\\frac{2}{x^{2}+1}-1" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\int\\:\\frac{2}{1+x^{2}}-1dx" }, { "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{2}{1+x^{2}}dx-\\int\\:1dx" }, { "type": "interim", "title": "$$\\int\\:\\frac{2}{1+x^{2}}dx=2\\arctan\\left(x\\right)$$", "input": "\\int\\:\\frac{2}{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": "=2\\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": "=2\\arctan\\left(x\\right)" } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "interim", "title": "$$\\int\\:1dx=x$$", "input": "\\int\\:1dx", "steps": [ { "type": "step", "primary": "Integral of a constant: $$\\int{a}dx=ax$$", "result": "=1\\cdot\\:x" }, { "type": "step", "primary": "Simplify", "result": "=x", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Integrals", "interimType": "Integrals" } }, { "type": "step", "result": "=2\\arctan\\left(x\\right)-x" }, { "type": "step", "primary": "Add a constant to the solution", "result": "=2\\arctan\\left(x\\right)-x+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=Sum%20Rule", "practiceTopic": "Integral Sum Rule" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "y=2\\arctan(x)-x+C" }, "showViewLarger": true } }, "meta": { "showVerify": true } }