2023TrialShort answer

Q12 (b) (ii)

1 mark

Question

By making the substitution $u=a-x$ , show that

\int_0^a f(x)\,dx=\int_0^a f(a-x)\,dx.

Try the question before revealing the answer.

Solution

(ii)

\int_0^a f(x)\,dx

Let $u=a-x$ , $x=a-u$ .

\frac{du}{dx}=-1

dx=-du

When $x=a$ , $u=0$ .

When $x=0$ , $u=a$ .

\begin{aligned} \int_0^a f(x)\,dx &=\int_a^0 f(a-u)(-du)\\ &=\int_0^a f(a-u)\,du \end{aligned}

As $u$ is a dummy variable,

\int_0^a f(x)\,dx=\int_0^a f(a-x)\,dx

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Syllabus alignment

Calculus

Further IntegrationApply substitution, partial fractions, integration by parts and recurrence relationships to theoretical and practical problems.

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