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

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
2
JEE Main 2025 (Online) 4th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The value of $\int_\limits{-1}^1 \frac{(1+\sqrt{|x|-x}) e^x+(\sqrt{|x|-x}) e^{-x}}{e^x+e^{-x}} d x$ is equal to

A
$1+\frac{2 \sqrt{2}}{3}$
B
$1-\frac{2 \sqrt{2}}{3}$
C
$2+\frac{2 \sqrt{2}}{3}$
D
$3-\frac{2 \sqrt{2}}{3}$
3
JEE Main 2025 (Online) 3rd April Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
The integral $\int_0^\pi \frac{8 x d x}{4 \cos ^2 x+\sin ^2 x}$ is equal to
A
$2 \pi^2$
B
$4 \pi^2$
C
$\pi^2$
D
$\frac{3 \pi^2}{2}$
4
JEE Main 2025 (Online) 3rd April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let the domain of the function $f(x)=\log _2 \log _4 \log _6\left(3+4 x-x^2\right)$ be $(a, b)$. If $\int_0^{b-a}\left[x^2\right] d x=p-\sqrt{q}-\sqrt{r}, p, q, r \in \mathbb{N}, \operatorname{gcd}(p, q, r)=1$, where $[\cdot]$ is the greatest integer function, then $p+q+r$ is equal to

A
10
B
11
C
9
D
8
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