If $f$ and $g$ are continuous functi

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Target Exam

CUET

Subject

-- Mathematics - Section A

Chapter

Definite Integration

Question:

If $f$ and $g$ are continuous function in [0,1] satisfying $f(x)=f(a-x)$ and $g(x)+g(a-x)=a$, then $\int_{0}^{a}f(x)g(x)dx=$

Options:

$\frac{a}{2}\int_{0}^{a}f(x)dx$

$\frac{a}{2}$

$\int_{0}^{a}f(x)dx$

$a\int_{0}^{a}f(x)dx$

Correct Answer:

$\frac{a}{2}\int_{0}^{a}f(x)dx$

Explanation:

$I = \int\limits_{0}^{a} f(x) g(x) dx$

$I = \int\limits_{0}^{a} f(a − x) g(a − x) dx$

$I = \int\limits_{0}^{a} f(x) g(a −g(x)) dx$

$I = a\int\limits_{0}^{a}f(x) dx − \int\limits_{0}^{a}f(x)g(x)dx$

$I = a\int\limits_{0}^{a}f(x)dx − I$

⇒ $2I = a\int\limits_{0}^{a}f(x) dx$

⇒$I = \frac{a}{2}\int \limits_{0}^{a}f(x)dx$