Integrals

Elementary integration methods

Look for a derivative hidden beside a composition, simplify algebraically, or use a trigonometric identity. Verify every primitive by differentiation.

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1. Undo the chain rule

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Example graph

f(x)=2x\cos(x^2),\quad F(x)=\sin(x^2)

Set u=x², so du=2x dx. Then ∫2x cos(x²) dx becomes ∫cos(u) du=sin(u)+C. The solid integrand is the derivative of the dashed primitive.

f(a): 1.080605
F(a): 0.841471

The formulas identify the solid and dashed curves. Compare their values using the slider; the accompanying explanation states their mathematical relationship.

2. Recognize u′/u

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Example graph

f(x)=\frac{2x}{1+x^2},\quad F(x)=\ln(1+x^2)

With u=1+x², du=2x dx. The primitive of 2x/(1+x²) is ln(1+x²)+C. Here u is positive everywhere, so no absolute-value ambiguity occurs.

f(a): 1
F(a): 0.693147

The formulas identify the solid and dashed curves. Compare their values using the slider; the accompanying explanation states their mathematical relationship.

3. Partial fractions

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Example graph

f=\frac1{x^2-1},\quad F=\frac12\ln\left|\frac{x-1}{x+1}\right|

Split 1/(x²−1) into 1/[2(x−1)]−1/[2(x+1)]. Integrating each term gives a logarithmic difference. Work on one interval avoiding ±1.

f(a): -1
F(a): 0

The formulas identify the solid and dashed curves. Compare their values using the slider; the accompanying explanation states their mathematical relationship.

4. A trigonometric identity

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Example graph

f=\sin^2x,\quad F=\frac x2-\frac{\sin2x}{4}

Use sin²x=(1−cos 2x)/2. Its primitive is x/2−sin(2x)/4+C; the factor 1/4 includes the inner derivative of 2x.

f(a): 0.708073
F(a): 0.272676

The formulas identify the solid and dashed curves. Compare their values using the slider; the accompanying explanation states their mathematical relationship.