JEE MAIN Definite Integration Previous Year Questions (PYQs) – Page 4 of 17
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Let \(f(x)\) be a function satisfying \(f(x)+f(\pi-x)=\pi^{2}\), \(\forall x\in\mathbb{R}\).
Then \(\displaystyle \int_{0}^{\pi} f(x)\sin x\,dx\) is equal to:
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The value of the definite integral $\int\limits_{\pi /24}^{5\pi /24} {{{dx} \over {1 + \root 3 \of {\tan 2x} }}} $ is :
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Solution
The value of $\int\limits_{{{ - 1} \over {\sqrt 2 }}}^{{1 \over {\sqrt 2 }}} {{{\left( {{{\left( {{{x + 1} \over {x - 1}}} \right)}^2} + {{\left( {{{x - 1} \over {x + 1}}} \right)}^2} - 2} \right)}^{{1 \over 2}}}dx} $ is :
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The area (in sq. units) of the region, given by the set $\{ (x,y) \in R \times R|x \ge 0,2{x^2} \le y \le 4 - 2x\} $
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The integral $\displaystyle \int_{\pi/6}^{\pi/4}\frac{dx}{\sin 2x\,(\tan^{5}x+\cot^{5}x)}$ equals :
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If the value of the integral $\int\limits_0^5 {{{x + [x]} \over {{e^{x - [x]}}}}dx = \alpha {e^{ - 1}} + \beta } $, where $\alpha$, $\beta$ $\in$ R, 5$\alpha$ + 6$\beta$ = 0, and [x] denotes the greatest integer less than or equal to x; then the value of ($\alpha$ + $\beta$)2 is equal to :
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Let $\beta(m,n)=\displaystyle\int_{0}^{1}x^{m-1}(1-x)^{,n-1},dx,; m,n>0$. If $\displaystyle\int_{0}^{1}(1-x^{10})^{20},dx=a\times \beta(b,c)$, then $100(a+b+c)$ equals:
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The value of the integral $\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {(1 + {e^x})({{\sin }^6}x + {{\cos }^6}x)}}} $ is equal to
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Solution
If $I_1=\displaystyle\int_{0}^{1} e^{-x}\cos^{2}x,dx$;
$I_2=\displaystyle\int_{0}^{1} e^{-x^{2}}\cos^{2}x,dx$ and
$I_3=\displaystyle\int_{0}^{1} e^{-x^{3}},dx$; then
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Solution
$ \text{Let } \alpha,\beta,\gamma \text{ be the three roots of } x^{3}+bx+c=0. \text{ If } \beta\gamma=1=-\alpha,\ \text{then } b^{3}+2c^{3}-3\alpha^{3}-6\beta^{3}-8\gamma^{3} \text{ is equal to:} $
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Solution
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