11.5PureStretch

Integration by Substitution

Substitution simplifies an integral by replacing a chunk of the integrand with a new variable u. You must change every part — the function, the dx, and (for definite integrals) the limits.

28 min Video by Zeeshan Zamurred Integration

Edexcel A level Maths: 11.5 Integration By Substitution (Part 1)Watch the full walkthrough 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 $\frac{d u}{d x}$ and rearrange for $d x$ . 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) d x = \int f (u) d u, u = g (x)

Substitution.

\frac{d u}{d x} = 2 x \Rightarrow d u = 2 x d x

\int u^{3} d u = \frac{u ^{4}}{4} + c

3Back-substitute:

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

Answer

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

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

Formula recap

u = g (x), d u = g^{'} (x) d x

Set up the substitution.

\int f (u) d u

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.

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