JEE MAIN Definite Integration Previous Year Questions (PYQs) – Page 1 of 17
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If I1 = $\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{100}}} dx$ andI2 = $\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{101}}} dx$ such that I2 = $\alpha $I1then $\alpha $ equals to :
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Solution
Let $a \in \left(0, \dfrac{\pi}{2}\right)$ be fixed.
If
$\displaystyle \int \dfrac{\tan x + \tan a}{\tan x - \tan a} , dx = A(x)\cos 2a + B(x)\sin 2a + C,$
where $C$ is a constant of integration,
then the functions $A(x)$ and $B(x)$ are respectively:
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Solution
If $I=\displaystyle\int_{0}^{\pi/2}\frac{\sin^{3/2}x}{\sin^{3/2}x+\cos^{3/2}x}\,dx$, then $\displaystyle\int_{0}^{2I}\frac{x\sin x\cos x}{\sin^{4}x+\cos^{4}x}\,dx$ equals:
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Solution
The value of the definite integral$ \int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{dx} \over {(1 + {e^{x\cos x}})({{\sin }^4}x + {{\cos }^4}x)}}} $ is equal to :
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Solution
Let $f(x)=\int_{0}^{x}\left(t+\sin(1-e^{t})\right)dt,\;x\in\mathbb{R}$.
Then, $\displaystyle\lim_{x\to0}\dfrac{f(x)}{x^{3}}$ is equal to:
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Solution
The value of $\displaystyle \int_{-\pi/2}^{\pi/2} \dfrac{dx}{[x] + [\sin x] + 4}$,
where $[t]$ denotes the greatest integer less than or equal to $t$, is :
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Solution
If the area of the bounded region $R = \left\{ {(x,y):\max \{ 0,{{\log }_e}x\} \le y \le {2^x},{1 \over 2} \le x \le 2} \right\}$ is , $\alpha {({\log _e}2)^{ - 1}} + \beta ({\log _e}2) + \gamma $, then the value of ${(\alpha + \beta - 2\lambda )^2}$ is equal to :
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Solution
$\int_{0}^{20\pi} (|\sin x| + |\cos x|)^{2} \, dx$ is equal to:
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Solution
The value of the integral $\displaystyle \int_{-\pi/4}^{\pi/4}\frac{x+\pi/4}{\,2-\cos 2x\,}\,dx$ is:
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Solution
$ \text{The integral } \int \dfrac{\left(1-\tfrac{1}{\sqrt{3}}\right)(\cos x-\sin x)}{1+\tfrac{2}{\sqrt{3}}\sin 2x},dx \text{ is equal to :}$
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Solution
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