JEE MAIN Definite Integration Previous Year Questions (PYQs) – Page 15 of 17
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Let $g(t) = \int_{ - \pi /2}^{\pi /2} {\cos \left( {{\pi \over 4}t + f(x)} \right)} dx$, where $f(x) = {\log _e}\left( {x + \sqrt {{x^2} + 1} } \right),x \in R$. Then which one of the following is correct?
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Solution
Let for $f(x)=7 \tan ^8 x+7 \tan ^6 x-3 \tan ^4 x-3 \tan ^2 x, \quad \mathrm{I}_1=\int_0^{\pi / 4} f(x) \mathrm{d} x$ and $\mathrm{I}_2=\int_0^{\pi / 4} x f(x) \mathrm{d} x$. Then $7 \mathrm{I}_1+12 \mathrm{I}_2$ is equal to :
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Solution
If $\int\limits_0^{100\pi } {{{{{\sin }^2}x} \over {{e^{\left( {{x \over \pi } - \left[ {{x \over \pi }} \right]} \right)}}}}dx = {{\alpha {\pi ^3}} \over {1 + 4{\pi ^2}}},\alpha \in R} $ where [x] is the greatest integer less than or equal to x, then the value of $\alpha$ is :
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If $\displaystyle \int_{0}^{\pi/3}\!\cos^{4}x\,dx=a\pi+b\sqrt{3}$, where $a$ and $b$ are rational numbers, then $9a+8b$ is equal to:
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Solution
Let $y=y(x)$ be the solution of the differential equation $\dfrac{dy}{dx}+3\tan^2 x,y+3y=\sec^2 x$, $y(0)=\dfrac{1}{3}+e^3$. Then $y!\left(\dfrac{\pi}{4}\right)$ is equal to:
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Let $\alpha\in(0,1)$ and $\beta=\log_{e}(1-\alpha)$. Let $P_{n}(x)=x+\dfrac{x^{2}}{2}+\dfrac{x^{3}}{3}+\cdots+\dfrac{x^{n}}{n},\ x\in(0,1)$. Then the integral $\displaystyle \int_{0}^{\alpha}\frac{t^{50}}{1-t}\,dt$ is equal to
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Solution
$\displaystyle \int_{\pi/6}^{\pi/3}\sec^{\tfrac{2}{3}}x;\csc^{\tfrac{4}{3}}x,dx$ is equal to:
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Solution
The value of the integral
$\int\limits_4^{10} {{{\left[ {{x^2}} \right]dx} \over {\left[ {{x^2} - 28x + 196} \right] + \left[ {{x^2}} \right]}}} ,$
where [x] denotes the greatest integer less than or equal to x, is :
$\int\limits_4^{10} {{{\left[ {{x^2}} \right]dx} \over {\left[ {{x^2} - 28x + 196} \right] + \left[ {{x^2}} \right]}}} ,$
where [x] denotes the greatest integer less than or equal to x, is :
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Solution
The value of $\int\limits_{{{ - \pi } \over 2}}^{{\pi \over 2}} {{1 \over {1 + {e^{\sin x}}}}dx} $ is:
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Solution
The integral $\displaystyle \int_{0}^{\pi}\frac{8x,dx}{4\cos^{2}x+\sin^{2}x}$ is equal to
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Solution
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