Definite Integration-JEE-Mains-PYQ’s

Mathematics Questions

Let $[.]$ denote the greatest integer function. Then
$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{12(3 + [x])}{3 + [\sin x] + [\cos x]} \right) dx$
is equal to :

(A) $12\pi + 5$

(B) $11\pi + 2$

(D) $13\pi + 1$

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Let $f$ be a polynomial function such that $f(x^2 + 1) = x^4 + 5x^2 + 2$ , for all $x \in \mathbb{R}$ . Then $\int_{0}^{3} f(x) dx$ is equal to

(A) $\frac{33}{2}$

(B) $\frac{5}{3}$

(D) $\frac{41}{3}$

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The value of the integral $\int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1 + \sqrt[3]{\tan 2x}}$ is :

(A) $\frac{\pi}{3}$

(B) $\frac{\pi}{18}$

(D) $\frac{\pi}{12}$

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The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{1}{[x] + 4} \right) dx$ , where $[.]$ denotes the greatest integer function, is

(A) $\frac{1}{60}(21\pi - 1)$

(B) $\frac{1}{60}(\pi - 7)$

(D) $\frac{7}{60}(3\pi - 1)$

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Let $f : [1, \infty) \to \mathbb{R}$ be a differentiable function. If $6 \int_{1}^{x} f(t) dt = 3xf(x) + x^3 - 4$ for all $x \ge 1$ , then the value of $f(2) - f(3)$ is :

(A) 4

(B) 3

(D) -3

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The value of $\int_{-\pi/6}^{\pi/6} \left( \frac{\pi + 4x^{11}}{1 - \sin(|x| + \pi/6)} \right) dx$ is equal to:

(A) $8\pi$

(B) $4\pi$

(D) $6\pi$

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The integral $\int_{-1}^{3} (\pi^2 x \sin(\pi x)) \, dx$ is equal to

(A) $2 + 3\pi$

(B) $4 + \pi$

(D) $3 + 2\pi$

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Let $f(x)$ be a positive function and $I_1 = \int_{-\frac{1}{2}}^{1} 2x \, f(2x(1 - 2x)) \, dx$ and $I_2 = \int_{-1}^{2} f(x(1 - x)) \, dx$ . Then the value of $\frac{I_2}{I_1}$ is equal to