1
JEE Main 2024 (Online) 1st February Evening Shift
+4
-1
If $\int\limits_0^{\frac{\pi}{3}} \cos ^4 x \mathrm{~d} x=\mathrm{a} \pi+\mathrm{b} \sqrt{3}$, where $\mathrm{a}$ and $\mathrm{b}$ are rational numbers, then $9 \mathrm{a}+8 \mathrm{b}$ is equal to :
A
2
B
1
C
3
D
$\frac{3}{2}$
2
JEE Main 2024 (Online) 1st February Evening Shift
+4
-1
The value of $\int\limits_0^1\left(2 x^3-3 x^2-x+1\right)^{\frac{1}{3}} \mathrm{~d} x$ is equal to :
A
-1
B
2
C
0
D
1
3
JEE Main 2024 (Online) 1st February Morning Shift
+4
-1
The value of the integral $\int\limits_0^{\pi / 4} \frac{x \mathrm{~d} x}{\sin ^4(2 x)+\cos ^4(2 x)}$ equals :
A
$\frac{\sqrt{2} \pi^2}{8}$
B
$\frac{\sqrt{2} \pi^2}{16}$
C
$\frac{\sqrt{2} \pi^2}{32}$
D
$\frac{\sqrt{2} \pi^2}{64}$
4
JEE Main 2024 (Online) 31st January Evening Shift
+4
-1

Let $$f, g:(0, \infty) \rightarrow \mathbb{R}$$ be two functions defined by $$f(x)=\int\limits_{-x}^x\left(|t|-t^2\right) e^{-t^2} d t$$ and $$g(x)=\int\limits_0^{x^2} t^{1 / 2} e^{-t} d t$$. Then, the value of $$9\left(f\left(\sqrt{\log _e 9}\right)+g\left(\sqrt{\log _e 9}\right)\right)$$ is equal to :

A
10
B
9
C
8
D
6
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