JEE MAIN Definite Integration Previous Year Questions (PYQs) – Page 7 of 17
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If the value of the integral
$\displaystyle \int_{-\pi/2}^{\pi/2}
\left(
\dfrac{x^{2}\cos x}{1+x^{2}}
+\dfrac{1+\sin^{2}x}{1+e^{\sin(2\tan^{-1}x)}}
\right)\,dx
= \dfrac{\pi}{4}\,(\pi+a)-2,$
then the value of $a$ is:
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2
3
4
Solution
Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function such that $f(2)=1$. If $F(x)=x f(x)$ for all $x\in\mathbb{R}$, $\displaystyle\int_{0}^{2} x F''(x),dx=6$ and $\displaystyle\int_{0}^{2} x^{2} F''(x),dx=40$, then $F'(2)+\displaystyle\int_{0}^{2} F(x),dx$ is equal to:
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2
3
4
Solution
Let A1 be the area of the region bounded by the curves y = sinx, y = cosx and y-axis in the first quadrant. Also, let A2 be the area of the region bounded by the curves y = sinx, y = cosx, x-axis and x = ${\pi \over 2}$ in the first quadrant. Then,
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4
Solution
If $[t]$ denotes the greatest integer $\leq t$, then the value of
$
\int_{0}^{1} \left[ 2x - |3x^{2} - 5x + 2| + 1 \right] \, dx
$
is :
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4
Solution
For
$
I(x) = \int \frac{\sec^{2}x - 2022}{\sin^{2022}x} \, dx,
$
if
$
I\!\left(\frac{\pi}{4}\right) = 2^{1011},
$
then
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4
Solution
Let $f$ be a continuous function satisfying
$\displaystyle \int_{0}^{t^2} \big(f(x) + x^2\big)\,dx = \dfrac{4}{3}t^3, \; \forall t > 0.$
Then $f\!\left(\dfrac{\pi^2}{4}\right)$ is equal to:
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4
Solution
The area bounded by the curve y = |x2 $-$ 9| and the line y = 3 is :
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Solution
If $\sin\!\left(\dfrac{y}{x}\right)=\log_e|x|+\dfrac{\alpha}{2}$
is the solution of the differential equation
$x\cos\!\left(\dfrac{y}{x}\right)\dfrac{dy}{dx}=y\cos\!\left(\dfrac{y}{x}\right)+x$
and $y(1)=\dfrac{\pi}{3}$, then $\alpha^{2}$ is equal to:
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4
Solution
If $f(x)=\displaystyle\int \frac{1}{x^{1/4}\left(1+x^{1/4}\right)},dx,; f(0)=-6$, then $f(1)$ is equal to:
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4
Solution
The value of the integral
\[
\int_{-\log_e 2}^{\log_e 2} e^x \left( \log_e\!\left( e^x + \sqrt{1 + e^{2x}} \right) \right) dx
\]
is equal to:
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2
3
4
Solution
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