2022TrialShort answer

Q14 (d) (i)

1 mark

Question

Use a suitable substitution to show that
$\int_0^a f(x)\,dx=\int_0^a f(a-x)\,dx.$

Try the question before revealing the answer.

Solution

Let $u=a-x$ .

$du=-dx$

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

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

$\text{RHS}=\int_0^a f(a-x)\,dx$
$=\int_a^0-f(u)\,du$
$=-\int_0^a-f(u)\,du$
$=\int_0^a f(x)\,dx$
$=\text{LHS}$

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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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