This page is under construction; the links to solutions do not work yet.
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Write the integrand as a sum of (negative) powers of x. | Solution |
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Substitution. | Solution |
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Substitute u = x + 4, then multiply out. | Solution |
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Substitute u = - x^2, then integrate by parts. | Solution |
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Complete the square, then use trigonometric substitution. | Solution |
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Substitute u for the integrand, then use partial fraction decomposition. |
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Note sech^2 in integrand; IBP. | Solution |
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IBP with u = sec^3; use trig. identity to reduce to sec^3 integral. IBP again with u = sec. |
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Substitute u = arcsin x. | Solution |
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Multiply numerator and denominator by (1 - cos x); expand numerator. |
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Trigonometric substitution; double-angle formulas. | Solution |
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Partial fraction decomposition. | Solution |
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Substitution. | Solution |
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Expand and integrate term by term. | Solution |
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Substitute u = x - 1; expand and integrate term by term. | Solution |
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IBP with u = x^2. | Solution |
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Substitute u = cos x; then IBP. | Solution |
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Complete the square, then use substitution. | Solution |
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Substitute u = x^(3/2); IBP. | Solution |
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Multiply numerator and denominator by (1 - sin x); expand numerator. |
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Substitute u = tan(x/2). | Solution |
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IBP twice; solve for original. | Solution |
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Trigonometric substitution. | Solution |
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Partial fraction decomposition. | Solution |
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Expand and integrate term by term. | Solution |
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Substitution. | Solution |
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Substitute u = x + 3; expand; integrate term by term. | Solution |
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IBP with u = x^2. | Solution |
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Substitute u = x^3; complete square; trigonometric substitution. |
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Trigonometric substitution. | Solution |
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Multiply numerator and denominator by (1 - cos x); expand; integrate term by term. |
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IBP with u = log x; partial fraction decomposition. | Solution |
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Partial fraction decomposition. | Solution |
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Substitute y = x + 1; split integrand into two terms; IBP one of them with u = exp; miraculous cancellation! |
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Substitute u = x^(1/2); partial fraction decomposition. | Solution |
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IBP with u = log(1 + x^2), then partial fraction decomposition. |
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Substitute u = sin x; use a trig. identity; expand; integrate term by term. |
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IBP twice; solve for original integral. | Solution |
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Expand; integrate term by term. | Solution |
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Substitute u = log x. | Solution |
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Substitute u = (x-1)^(1/2); then trig. substitution. | Solution |
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Substitute u = x^(1/2); IBP. | Solution |
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Substitute u = x^4; partial fraction decomposition. IBP. | Solution |
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Multiply numerator and denominator by (sec x - 1); expand; integrate term by term. |
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IBP with u = (arcsin x)^2; substitute y = arcsin x. | Solution |
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Substitute x = sin t; then substitute u = tan(t/2). | Solution |
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Substitute x = u^6; use long division. | Solution |
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Long division. | Solution |
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Use trig. identity, then IBP with u = x. | Solution |
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IBP with u = log x. | Solution |
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Substitute u = sin x; cancel factors. | Solution |
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Substitute u = e^x. | Solution |
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Substitute u = tan x; partial fraction decomposition. | Solution |
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Separate integrand into 2 terms. IBP each: One with u = x; other with u = exp(x^2). |
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Multiply numerator and denominator by (x-1)^(1/2); substitute x = sec t. |
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e^2 is a constant. | Solution |
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Use trig. identity. | Solution |
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IBP with u = arctan x. | Solution |
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Partial fraction decomposition. | Solution |
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Substitute u = 1 + x^(3/2). | Solution |
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Write in terms of sin and cos; separate into 2 terms; IBP each, one with u = x; other with u = (sin x)^(1/2); miraculous cancellation! |
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Substitute u = (x^2 + x^-2)^(1/2). | Solution |
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IBP with u equal to the integrand. | Solution |
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Substitute u = log x; IBP twice. | Solution |
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Simplify the integrand. | Solution |
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IBP with u equal to the integrand. | Solution |
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IBP with u = x. | Solution |
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Partial fraction decomposition. | Solution |
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Substitute u = sin x; factor u from denominator; partial-fraction-like decomposition. |
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Multiply numerator and denominator by (x-1)^(1/2); expand; integrate each term. |
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Long division. | Solution |
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Watch the sign! | Solution |
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IBP with u = x. | Solution |
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Substitute u = sin x. | Solution |
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Integrate each term. | Solution |
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Substitute u = e^x. | Solution |
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Substitute u = log t. | Solution |
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Partial fraction decomposition. | Solution |
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"Rationalize" the denominator. | Solution |
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Multiply numerator and denominator by (1 - sin x); expand; integrate term by term. |
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Integrate each term. | Solution |
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Partial fraction decomposition by inspection. | Solution |
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Expand and integrate each term. | Solution |
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Trigonometric substitution. | Solution |
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Substitute u = (x-7)^(1/2). | Solution |
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Complete the square; substitute u=x+1. | Solution |
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Complete the square; substitute u=x+1. | Solution |
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Integrate by parts. | Solution |
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Trigonometric identity. | Solution |
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Substitute x = u^2; trigonometric substitution. | Solution |
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Trigonometric identity. | Solution |
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If you haven't memorized it, try substituting u = cos x. | Solution |
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Substitute u = sin x. | Solution |
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Substitute u = e^x; partial fraction decomposition. | Solution |
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Substitute u = e^x; partial fraction decomposition. | Solution |
This page is under construction; the links to solutions do not work yet.