1
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
$\frac{\sqrt{2} \pi^2}{64}$
2
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
3
JEE Main 2024 (Online) 30th January Evening Shift
+4
-1

Let $$f: \mathbb{R} \rightarrow \mathbb{R}$$ be a function defined by $$f(x)=\frac{x}{\left(1+x^4\right)^{1 / 4}}$$, and $$g(x)=f(f(f(f(x))))$$. Then, $$18 \int_0^{\sqrt{2 \sqrt{5}}} x^2 g(x) d x$$ is equal to

A
36
B
33
C
39
D
42
4
JEE Main 2024 (Online) 30th January Evening Shift
+4
-1

Let $$y=f(x)$$ be a thrice differentiable function in $$(-5,5)$$. Let the tangents to the curve $$y=f(x)$$ at $$(1, f(1))$$ and $$(3, f(3))$$ make angles $$\pi / 6$$ and $$\pi / 4$$, respectively with positive $$x$$-axis. If $$27 \int_\limits1^3\left(\left(f^{\prime}(t)\right)^2+1\right) f^{\prime \prime}(t) d t=\alpha+\beta \sqrt{3}$$ where $$\alpha, \beta$$ are integers, then the value of $$\alpha+\beta$$ equals

A
26
B
$$-$$16
C
36
D
$$-$$14
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