11.5PureStretch

Integration by Substitution

Learn integration by substitution with worked examples, including changed limits and integrals where x must also be rewritten. Replace the function and dx consistently, integrate in u, then check by differentiation.

28 min Integration

Edexcel A level Maths: 11.5 Integration By Substitution (Part 1)Curated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Choose a suitable substitution u
Replace dx using du = (du/dx)dx
Change the limits for definite integrals
Integrate and substitute back

The method

Let $u$ be the awkward inner part. Find $\dfrac{du}{dx}$ and rearrange for $dx$ . Rewrite the whole integral in terms of $u$ , integrate, then substitute back (or change the limits for a definite integral).

\int f(g(x))g'(x)\,dx = \int f(u)\,du, \quad u = g(x)

Substitution.

Example 1Find

\int 2x(x^2 + 1)^3\,dx

using

u = x^2 + 1

\tfrac{du}{dx} = 2x \Rightarrow du = 2x\,dx

\int u^3\,du = \tfrac{u^4}{4} + c

3Back-substitute:

\tfrac{(x^2+1)^4}{4} + c

Answer

\tfrac{(x^2+1)^4}{4} + c

Tip — For definite integrals, change the limits to u-values rather than back-substituting.

Changing the limits: a definite integral

Convert each original x-limit using the same substitution as the integrand. Once everything is in u, evaluate with u-limits. Alternatively, substitute back to x before applying the original limits; never mix the two variables.

Example 1Evaluate

\int_0^1 2x(x^2+1)^3\,dx

1Set

u=x^2+1

, so

du=2x\,dx

2When

x=0

u=1

; when

x=1

u=2

\int_1^2 u^3\,du=[\tfrac{u^4}{4}]_1^2=\tfrac{16}{4}-\tfrac{1}{4}

Answer

\dfrac{15}{4}

Tip — The antiderivative u^4/4 evaluated at 0 and 1 would answer a different integral. Label the new limits before integrating.

When an x remains after substitution

Sometimes du replaces dx but an extra factor of x remains. Rearrange the substitution to express that factor in u too. Do not integrate an expression containing independent x and u variables.

Example 1Find

\int x\sqrt{x+1}\,dx

using

u=x+1

du=dx

, but the factor

x

is still present. From

u=x+1

, write

x=u-1

\int (u-1)u^{1/2}\,du=\int(u^{3/2}-u^{1/2})\,du

3Integrate each power:

\tfrac25u^{5/2}-\tfrac23u^{3/2}+c

4Substitute

u=x+1

back into both terms. For real values this example uses

x\geq-1

5Differentiate to check:

(x+1)^{3/2}-(x+1)^{1/2}=x\sqrt{x+1}

Answer

\dfrac25(x+1)^{5/2}-\dfrac23(x+1)^{3/2}+c

Formula recap

u = g(x), \ du = g'(x)\,dx

Set up the substitution.

\int f(u)\,du

Integral in u.

Common mistakes to avoid

Leaving some x’s in the u-integral.

Convert everything (including dx) to u.

Keeping the original x-limits after substituting.

Change limits to u-values for a definite integral.

Key takeaways

Pick u, find du = g′(x)dx, rewrite fully in u.
Integrate, then substitute back (or change limits).
No x should remain in the u-integral.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

For $u=x^2+1$ , what are the new limits when x runs from 1 to 3?
- 2 to 10
- 1 to 3
- 1 to 9
- 0 to 8
Show answer and explanation
Answer: 2 to 10
Substitute each endpoint into u=x^2+1: u(1)=2 and u(3)=10. These are u-limits, not the original x-limits.
In a substitution you must also convert:
- dx into du
- only the function
- only the limits
- nothing else
Show answer and explanation
Answer: dx into du
Whole integrand.
For $\int 2x(x^2+1)^3 dx$ , a good substitution is:
- $u = x^2+1$
- $u = 2x$
- $u = x$
- $u = x^3$
Show answer and explanation
Answer: $u = x^2+1$
Its derivative is 2x.
That integral evaluates to:
- $\tfrac{(x^2+1)^4}{4}+c$
- $(x^2+1)^4 + c$
- $\tfrac{(x^2+1)^3}{3}+c$
- $2x(x^2+1)^4 + c$
Show answer and explanation
Answer: $\tfrac{(x^2+1)^4}{4}+c$
∫u³du.
For a definite integral it is best to:
- change the limits to u
- keep x-limits
- ignore limits
- use degrees
Show answer and explanation
Answer: change the limits to u
Avoids back-substituting.
After the u-integral, no remaining variable should be:
- x
- u
- c
- du
Show answer and explanation
Answer: x
Everything in u.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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