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Integration by substitution

Use uu-substitution (the reverse chain rule) to evaluate indefinite and definite integrals; change limits correctly when applying substitution to definite integrals.

Worked examples

Indefinite integral (power function)

Straightforward

Problem

Find ∫3x2(x3−1)5 dx\displaystyle\int 3x^2(x^3 - 1)^5\,dx.

Definite integral (changing limits)

Moderate

Problem

Evaluate ∫02x(x2+4)2 dx\displaystyle\int_0^2 \frac{x}{(x^2+4)^2}\,dx.

Trigonometric integrand

Challenging

Problem

Evaluate ∫0πsin⁡3xcos⁡x dx\displaystyle\int_0^{\pi} \sin^3 x \cos x\,dx.

Practise

Q1·Straightforward
To evaluate ∫2x(x2+3)4 dx\displaystyle\int 2x(x^2+3)^4\,dx using the substitution u=x2+3u = x^2 + 3, what is dudu in terms of dxdx?
Q2·Straightforward
Using the substitution u=3x+1u = 3x + 1, evaluate ∫(3x+1)5 dx\displaystyle\int (3x+1)^5\,dx. Which of the following is correct?
Q3·Straightforward
Evaluate ∫012x(x2+1)3 dx\displaystyle\int_0^1 2x(x^2+1)^3\,dx. Give your answer as a fraction in the form p/qp/q (enter the decimal equivalent to 2 decimal places).
Q4·Straightforward
Using the substitution u=x3−2u = x^3 - 2, the integral ∫3x2ex3−2 dx\displaystyle\int 3x^2 e^{x^3 - 2}\,dx becomes:
Q5·Moderate
Using the substitution u=sin⁡xu = \sin x, find ∫cos⁡x⋅sin⁡4x dx\displaystyle\int \cos x \cdot \sin^4 x\,dx in the form sin⁡nxk+C\dfrac{\sin^n x}{k} + C. What is the value of kk?
Q6·Moderate
Evaluate ∫1eln⁡xx dx\displaystyle\int_1^e \frac{\ln x}{x}\,dx. Give your answer to 2 decimal places.
Q7·Moderate
Use the substitution u=4−x2u = 4 - x^2 to find ∫01x4−x2 dx\displaystyle\int_0^1 \frac{x}{\sqrt{4-x^2}}\,dx. Give your answer to 4 decimal places.
Q8·Moderate
Find ∫xx+1 dx\displaystyle\int x\sqrt{x+1}\,dx using the substitution u=x+1u = x + 1. The answer has the form 2(x+1)3/2(ax+b)15+C\dfrac{2(x+1)^{3/2}(ax+b)}{15} + C. What is the value of aa?
Q9·Moderate
Evaluate ∫0π/4sin⁡(2x) dx\displaystyle\int_0^{\pi/4} \sin(2x)\,dx. Give your answer to 4 decimal places.
Q10·Challenging
Use the substitution u=ex+1u = e^x + 1 to evaluate ∫0ln⁡3ex(ex+1)2 dx\displaystyle\int_0^{\ln 3} \frac{e^x}{(e^x+1)^2}\,dx. Give your answer as a fraction (enter the decimal to 4 decimal places).
Q11·Challenging
A student claims that ∫02xx2+1 dx=ln⁡52\displaystyle\int_0^2 \frac{x}{x^2+1}\,dx = \dfrac{\ln 5}{2}. Use the substitution u=x2+1u = x^2 + 1 to verify this. What is the value of ln⁡52\dfrac{\ln 5}{2} to 4 decimal places?
Q12·Challenging
Use the substitution u=cos⁡xu = \cos x to evaluate ∫0π/2sin⁡xcos⁡3x dx\displaystyle\int_0^{\pi/2} \sin x \cos^3 x\,dx. Give your answer to 4 decimal places.