JEE MAIN Definite Integration Previous Year Questions (PYQs) – Page 10 of 17
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$\int\limits_{ - \pi }^\pi {\left| {\pi - \left| x \right|} \right|dx} $ is equal to :
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Solution
Let $\int_\limits\alpha^{\log _e 4} \frac{\mathrm{d} x}{\sqrt{\mathrm{e}^x-1}}=\frac{\pi}{6}$. Then $\mathrm{e}^\alpha$ and $\mathrm{e}^{-\alpha}$ are the roots of the equation :
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Solution
If
$\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\dfrac{x^2\cos x}{1+x^2}+\dfrac{1+\sin^2 x}{1+e^{\sin 2x}}\right)dx = \dfrac{\pi}{4}(\pi+a)-2$,
then the value of $a$ is:
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Solution
$a$ and $b$ be real constants such that the function $f$ defined by $f(x)=\left\{\begin{array}{ll}x^2+3 x+a & , x \leq 1 \\ b x+2 & , x>1\end{array}\right.$ be differentiable on $\mathbb{R}$. Then, the value of $\int_\limits{-2}^2 f(x) d x$ equals
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Solution
The integral $\int_{{\pi \over {12}}}^{{\pi \over 4}} {\,\,{{8\cos 2x} \over {{{\left( {\tan x + \cot x} \right)}^3}}}} \,dx$ equals :
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Solution
Let $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $f(x)=a e^{2 x}+b e^x+c x$. If $f(0)=-1, f^{\prime}\left(\log _e 2\right)=21$ and $\int_0^{\log _e 4}(f(x)-c x) d x=\frac{39}{2}$, then the value of $|a+b+c|$
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Solution
Let f : R $ \to $ R be defined as f(x) = e$-$xsinx. If F : [0, 1] $ \to $ R is a differentiable function with that F(x) = $\int_0^x {f(t)dt} $, then the value of $\int_0^1 {(F'(x) + f(x)){e^x}dx} $ lies in the interval
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Solution
$ \text{If } \displaystyle \int_{0}^{10}\frac{[\sin 2\pi x]}{e^{,x-[x]}},dx ;=; \alpha e^{-1}+\beta e^{-1/2}+\gamma,\ \text{ where } \alpha,\beta,\gamma \text{ are integers and } [x] \text{ is the greatest integer } \le x,\ \text{then the value of } \alpha+\beta+\gamma \text{ is:} $
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Solution
Let [t] denote the greatest integer less than or equal to t. Then, the value of the integral $\int\limits_0^1 {[ - 8{x^2} + 6x - 1]dx} $ is equal to :
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Solution
Evaluate the integral $ \displaystyle \int_{0}^{\infty}\frac{6}{e^{3x}+6e^{2x}+11e^{x}+6},dx $ is equal to:
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Solution
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