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STD 12 - 7.2 definite integral — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 7.2 definite integral1 Mark
MCQ
Let $f(x)$ be a function satisfying $f(x)+f(\pi-x)=$ $\pi^2, \forall x \in R$. Then $\int \limits_0^\pi f(x) \sin x d x$ is equal to $...........$.
  • A
    $\frac{\pi^2}{4}$
  • B
    $\frac{\pi^2}{2}$
  • C
    $2 \pi^2$
  • ✓
    $\pi^2$
✓

Answer

Correct option: D.
$\pi^2$
d
$f(x)+f(\pi-x)=\pi^2$

$I=\int \limits_0^\pi f(x) \sin x d x$

Applying King's Rule

$I=\int \limits_0^\pi f(\pi-x) \cdot \sin (\pi-x) d x$

$2 I=\int \limits_0^\pi[f(x)+f(\pi-x)] \sin x d x$

$2 I=\int \limits_0^\pi \pi^2 \sin x d x$

$2 I=\pi^2 \cdot \int \limits_0^\pi \sin x d x$

$2 I=\pi^2 \times 2$

$I=\pi^2$

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