1
JEE Main 2025 (Online) 8th April Evening Shift
MCQ (Single Correct Answer)
+4
-1

The integral $\int\limits_{-1}^{\frac{3}{2}} \left(| \pi^2 x \sin(\pi x) \right|) dx$ is equal to:

A

$2 + 3\pi$

B

$4 + \pi$

C

$1 + 3\pi$

D

$3 + 2\pi$

2
JEE Main 2025 (Online) 8th April Evening Shift
MCQ (Single Correct Answer)
+4
-1

Let f(x) be a positive function and $I_{1} = \int\limits_{-\frac{1}{2}}^{1} 2x \, f(2x(1-2x)) \, dx$ and $I_{2} = \int\limits_{-1}^{2} f(x(1-x)) \, dx$. Then the value of $\frac{I_{2}}{I_{1}}$ is equal to ________

A

12

B

9

C

6

D

4

3
JEE Main 2025 (Online) 7th April Morning Shift
MCQ (Single Correct Answer)
+4
-1

The integral $\int_0^\pi \frac{(x+3) \sin x}{1+3 \cos ^2 x} d x$ is equal to

A
$\frac{\pi}{\sqrt{3}}(\pi+1)$
B
$\frac{\pi}{3 \sqrt{3}}(\pi+6)$
C
$\frac{\pi}{\sqrt{3}}(\pi+2)$
D
$\frac{\pi}{2 \sqrt{3}}(\pi+4)$
4
JEE Main 2025 (Online) 4th April Evening Shift
MCQ (Single Correct Answer)
+4
-1

Let $f(x)+2 f\left(\frac{1}{x}\right)=x^2+5$ and $2 g(x)-3 g\left(\frac{1}{2}\right)=x, x>0$. If $\alpha=\int_1^2 f(x) \mathrm{d} x$, and $\beta=\int_1^2 g(x) \mathrm{d} x$, then the value of $9 \alpha+\beta$ is :

A
0
B
10
C
1
D
11
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