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# Prove that: ∫2a0f(x)dx=∫a0f(x)dx+∫a0f(2a−x)dx - HSC Arts 12th Board Exam - Mathematics and Statistics

ConceptProperties of Definite Integrals

#### Question

Prove that: int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2a-x)dx

#### Solution

LHS=int_0^(2a)f(x)dx=int_0^af(x)dx+int_a^(2a)f(x)dx.........(1)

Substitute x = a + t in the second integral

dx=dt

When x = a, t = 0.

When x = 2a, t = a.

thereforeint_a^(2a)f(x)dx=int_0^af(a+t)dt

=int_0^af(a+(a-t))dt (therefore int_0^af(x)dx=int_0^a f(a-x)dx)

=int_0^af(2a-t)dt

int_a^(2a)f(x)dx=int_0^af(2a-x)dx (therefore int_0^af(t)dt=int_0^af(x)dx)

Using the above in (1), we get

int_0^(2a)f(x)dx=int_0^af(x)dx+int_a^(2a)f(x)dx

=int_0^af(x)dx+int_0^af(2a-x)dx=RHS ("Proved")

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#### APPEARS IN

2015-2016 (March) (with solutions)
Question 6.1.1 | 3.00 marks

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Solution Prove that: ∫2a0f(x)dx=∫a0f(x)dx+∫a0f(2a−x)dx Concept: Properties of Definite Integrals.
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