Let $\mathrm{f}(\mathrm{x})$ be a function satisfying $\mathrm{f}(\mathrm{x})+\mathrm{f}(\pi-\mathrm{x})= \pi^{2}, \forall \mathrm{x} \in \mathrm{R}$. Then $\int_{0}^{\pi} \mathrm{f}(\mathrm{x}) \sin \mathrm{x} \mathrm{dx}$ is equal to

JEE Main · 2023 · 06 Apr 2023 · Shift 2 · Definite Integrals

Let

f (x)

be a function satisfying

f (x) + f (π - x) = π^{2}, \forall x \in R

. Then

\int_{0}^{π} f (x) sin x dx

is equal to

Answer: (D)

Let

I = \int_{0}^{π} f (x) sin x d x

. Using the King's property

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

, we have

I = \int_{0}^{π} f (π - x) sin (π - x) d x = \int_{0}^{π} f (π - x) sin x d x

. Adding the two expressions:

2 I = \int_{0}^{π} [f (x) + f (π - x)] sin x d x = \int_{0}^{π} π^{2} sin x d x = π^{2} [- cos x]_{0}^{π} = π^{2} (2)

. Therefore

I = π^{2}

JEE Main · 2023 · 06 Apr 2023 · Shift 2 · Single correct option

The value of $\int_{- π /6}^{π /6} (\frac{π + 4 x ^{11}}{1 - s i n ( ∣ x ∣ + π /6 )}) d x$ is equal to
JEE Main 2026View solution →
$6 \int_{0}^{π} ∣ (sin 3 x + sin 2 x + sin x) ∣ d x$ is equal to.......
JEE Main 2026View solution →
Let $[\cdot]$ denote the greatest integer function and $f (x) = lim_{n \to \infty} \frac{1}{n ^{3}} \sum_{k = 1}^{n} [\frac{k ^{2}}{3 ^{x}}]$ . Then $12 \sum_{j = 1}^{\infty} f (j)$ is equal …
JEE Main 2026View solution →
If $\int_{0}^{1} 4 cot^{- 1} (1 - 2 x + 4 x^{2}) dx = a tan^{- 1} (2) - b lo g_{e} (5)$ , where $a, b \in N$ , then $(2 a + b)$ is equal to ____.
JEE Main 2026View solution →
Let $f : [1, \infty) \to R$ be a differentiable function. If $6 \int_{1}^{x} f (t) d t = 3 x f (x) + x^{3} - 4$ for all $x \geq 1$ , then the value of $f (2) - f (3)$ is
JEE Main 2026View solution →
The value of $\int_{- \frac{π}{2}}^{\frac{π}{2}} (\frac{1}{[ x ] + 4}) dx$ , where [•] denotes the greatest integer function, is
JEE Main 2026View solution →