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Integration techniques

Integrate using integration by parts (∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du); decompose rational functions using partial fractions including repeated and irreducible quadratic factors; apply the t=tan⁡(x/2)t = \tan(x/2) Weierstrass substitution; evaluate trigonometric integrals and powers.

Worked examples

Integration by parts — choosing uu and dvdv

Straightforward

Problem

Evaluate ∫01x2ex dx\displaystyle\int_0^1 x^2 e^x\,dx.

Partial fractions with a repeated linear factor

Moderate

Problem

Evaluate ∫121x(x+1)2 dx\displaystyle\int_1^2 \frac{1}{x(x+1)^2}\,dx.

The Weierstrass (tt-)substitution

Challenging

Problem

Evaluate ∫0π/212+cos⁡x dx\displaystyle\int_0^{\pi/2} \frac{1}{2 + \cos x}\,dx using t=tan⁡ ⁣(x2)t = \tan\!\left(\tfrac{x}{2}\right).

Practise

Q1·Straightforward
Use integration by parts to evaluate ∫01xex dx\displaystyle\int_0^1 x e^x\,dx. (Let u=xu = x and dv=ex dxdv = e^x\,dx.)
Q2·Straightforward
Use integration by parts to evaluate ∫0π/2xcos⁡x dx\displaystyle\int_0^{\pi/2} x\cos x\,dx. Give your answer to two decimal places.
Q3·Straightforward
Evaluate ∫1eln⁡x dx\displaystyle\int_1^e \ln x\,dx using integration by parts. (Let u=ln⁡xu = \ln x and dv=dxdv = dx.)
Q4·Straightforward
Evaluate ∫01x2ex dx\displaystyle\int_0^1 x^2 e^x\,dx by applying integration by parts twice. Give your answer to two decimal places.
Q5·Moderate
Use partial fractions to evaluate ∫011(x+1)(x+2) dx\displaystyle\int_0^1 \frac{1}{(x+1)(x+2)}\,dx. Give your answer to two decimal places.
Q6·Moderate
Evaluate ∫232x+1x2+x dx\displaystyle\int_2^3 \frac{2x+1}{x^2+x}\,dx. Give your answer to two decimal places.
Q7·Moderate
Evaluate ∫0π/4tan⁡2x dx\displaystyle\int_0^{\pi/4} \tan^2 x\,dx using the identity tan⁡2x=sec⁡2x−1\tan^2 x = \sec^2 x - 1. Give your answer to two decimal places.
Q8·Moderate
Evaluate ∫0π/2sin⁡2x dx\displaystyle\int_0^{\pi/2} \sin^2 x\,dx using the double-angle identity sin⁡2x=1−cos⁡2x2\sin^2 x = \dfrac{1 - \cos 2x}{2}. Give your answer to two decimal places.
Q9·Moderate
Use partial fractions to evaluate ∫121x(x+1)2 dx\displaystyle\int_1^2 \frac{1}{x(x+1)^2}\,dx. Give your answer to two decimal places.
Q10·Challenging
Use the Weierstrass substitution t=tan⁡ ⁣(x2)t = \tan\!\left(\dfrac{x}{2}\right) to evaluate ∫0π/211+sin⁡x dx\displaystyle\int_0^{\pi/2} \frac{1}{1 + \sin x}\,dx.

Recall: sin⁡x=2t1+t2\sin x = \dfrac{2t}{1+t^2} and dx=21+t2 dtdx = \dfrac{2}{1+t^2}\,dt.
Q11·Challenging
Evaluate ∫0π/2cos⁡3x dx\displaystyle\int_0^{\pi/2} \cos^3 x\,dx by writing cos⁡3x=cos⁡x(1−sin⁡2x)\cos^3 x = \cos x(1 - \sin^2 x) and using the substitution u=sin⁡xu = \sin x. Express your answer as a fraction.
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Q12·Challenging
Evaluate ∫0πexsin⁡x dx\displaystyle\int_0^{\pi} e^x \sin x\,dx using integration by parts twice. (Hint: after two applications, you will find the original integral appears on both sides of the equation.) Give your answer to two decimal places.