JEE MAIN Definite Integration Previous Year Questions (PYQs) – Page 8 of 17
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If $\int {{1 \over x}\sqrt {{{1 - x} \over {1 + x}}} dx = g(x) + c} $, $g(1) = 0$, then $g\left( {{1 \over 2}} \right)$ is equal to :
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If $\displaystyle \int \frac{1}{a^{2}\sin^{2}x+b^{2}\cos^{2}x},dx=\frac{1}{12}\tan^{-1}(3\tan x)+\text{constant}$, then the maximum value of $a\sin x+b\cos x$ is:
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Solution
The value of $k\in\mathbb{N}$ for which the integral $I_n=\displaystyle\int_{0}^{1}(1-x^{k})^{n},dx,\ n\in\mathbb{N}$, satisfies $147I_{20}=148I_{21}$ is:
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If $f(x)=\dfrac{2-x\cos x}{2+x\cos x}$ and $g(x)=\log_e x,\ (x>0)$, then the value of the integral $\displaystyle \int_{-\pi/4}^{\pi/4} g\big(f(x)\big),dx$ is:
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The integral $\displaystyle \int_{-1}^{\tfrac{3}{2}} \left( |\pi^2 x \sin(\pi x)| \right) dx$ is equal to
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Solution
Consider the integral $I = \int_0^{10} {{{[x]{e^{[x]}}} \over {{e^{x - 1}}}}dx} $, where [x] denotes the greatest integer less than or equal to x. Then the value of I is equal to :
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Solution
Let $f(x)$ be a positive function and $I_{1}=\int_{-\tfrac{1}{2}}^{1} 2x,f\left(2x(1-2x)\right),dx$ and $I_{2}=\int_{-1}^{2} f\left(x(1-x)\right),dx$. Then the value of $\dfrac{I_{2}}{I_{1}}$ is equal to
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The integral $\displaystyle \int_{\pi/4}^{3\pi/4}\dfrac{dx}{1+\cos x}$ is equal to :
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The value of the integral $\int\limits_{ - 2}^2 {{{|{x^3} + x|} \over {({e^{x|x|}} + 1)}}dx} $ is equal to :
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Solution
If $f:\mathbb{R}\to\mathbb{R}$ is a continuous function satisfying
\[
\int_{0}^{\pi/2} f(\sin 2x)\,\sin x\,dx \;+\; \alpha \int_{0}^{\pi/4} f(\cos 2x)\,\cos x\,dx \;=\; 0,
\]
then the value of $\alpha$ is:
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Solution
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