JEE Main · 2019 · 12 Jan 2019 · Shift 1 · Definite Integrals

Let

f

and

g

be continuous functions on

[0, a]

such that

f (x) = f (a - x)

and

g (x) + g (a - x) = 4

, then

\int_{0}^{a} f (x) g (x) d x

is equal to:

Answer: (C)

Solution

Let

I = \int_{0}^{a} f (x) g (x) d x

and replace

x

a - x

I = \int_{0}^{a} f (a - x) g (a - x) d x

. Since

f (a - x) = f (x)

and

g (a - x) = 4 - g (x)

, this becomes

I = \int_{0}^{a} f (x) (4 - g (x)) d x = 4 \int_{0}^{a} f (x) d x - I

. Bringing the

I

across,

2 I = 4 \int_{0}^{a} f (x) d x

, so

I = 2 \int_{0}^{a} f (x) d x

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