Indefinite Integration-JEE-Mains-PYQ’s

Mathematics Questions

Let $f(x) = \int \frac{dx}{x^{2/3} + 2x^{1/2}}$ , be such that $f(0) = -26 + 24 \log_e(2)$ . If $f(1) = a + b \log_e(3)$ , where $a, b \in \mathbb{Z}$ , then $a + b$ is equal to:

(A) -5

(B) -11

(D) -26

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If $\int \left( \frac{1 - 5 \cos^2 x}{\sin^5 x \cos^2 x} \right) dx = f(x) + C$ , where $C$ is the constant of integration, then $f\left(\frac{\pi}{6}\right) - f\left(\frac{\pi}{4}\right)$ is equal to:

(A) $\frac{1}{\sqrt{3}}(26 - \sqrt{3})$

(B) $\frac{4}{\sqrt{3}}(8 - \sqrt{6})$

(D) $\frac{2}{\sqrt{3}}(4 + \sqrt{6})$

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Let $f(t) = \int \left( \frac{1 - \sin(\log_e t)}{1 - \cos(\log_e t)} \right) dt, t > 1$ . If $f(e^{\pi/2}) = -e^{\pi/2}$ and $f(e^{\pi/4}) = \alpha e^{\pi/4}$ , then $\alpha$ equals:

(A) $1 + \sqrt{2}$

(B) $-1 - 2\sqrt{2}$

(D) $-1 + \sqrt{2}$

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Let $I(x) = \int \frac{3dx}{(4x+6)\left(\sqrt{4x^2+8x+3}\right)}$ and $I(0) = \frac{\sqrt{3}}{4} + 20$ . If $I\left(\frac{1}{2}\right) = \frac{a\sqrt{2}}{b} + c$ , where $a, b, c \in \mathbb{N}, \text{gcd}(a,b) = 1$ , then $a + b + c$ is equal to:

(A) 30

(B) 29

(D) 31

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Let $f(x) = \int \frac{(2 - x^2) \cdot e^x}{(\sqrt{1+x})(1-x)^{3/2}} dx$ . If $f(0) = 0$ , then $f\left(\frac{1}{2}\right)$ is equal to:

(A) $\sqrt{2e} - 1$

(B) $\sqrt{2e} + 1$

(D) $\sqrt{3e} + 1$

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Let $f(x) = \int x^3 \sqrt{3 - x^2} \, dx$ . If $f(5/\sqrt{2}) = -4$ , then $f(1)$ is equal to

(A) $-\frac{6\sqrt{2}}{5}$

(B) $-\frac{8\sqrt{2}}{5}$

(D) $-\frac{4\sqrt{2}}{5}$

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If $f(x) = \int \frac{1}{x^{2/4}(1+x^{2/4})} \, dx$ , $f(0) = -6$ , then $f(1)$ is equal to :

(A) $4(\log_e 2 - 2)$

(B) $\log_e 2 + 2$

(D) $4(\log_e 2 + 2)$

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Let $\int x^3 \sin x \, dx = g(x) + C$ , where $C$ is the constant of integration. If $8 \left( g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right) \right) = \alpha \pi^3 + \beta \pi^2 + \gamma$ , $\alpha, \beta, \gamma \in \mathbb{Z}$ , then $\alpha + \beta - \gamma$ equals :

(A) 47

(B) 55

(D) 48

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Let $I(x) = \int \frac{dx}{(x-11)^5 (x+15)^{15}}$ . If $I(37) - I(24) = \frac{1}{4} \left( \frac{1}{b} - \frac{1}{c} \right)$ , $b, c \in \mathbb{N}$ , then $3(b + c)$ is equal to

(A) 39

(B) 22

(D) 26

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If $\int e^x \left( \frac{x \sin^{-1} x}{\sqrt{1-x^2}} + \frac{\sin^{-1} x}{(1-x^2)^{1/2}} + \frac{x}{1-x^2} \right) dx = g(x) + C$ , where $C$ is the constant of integration, then $g\left(\frac{1}{2}\right)$ equals :