JEE Main 2024 (Online) 1st February Evening Shift | Definite Integration Question 18 | Mathematics

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 :

$\frac{3}{2}$

JEE Main 2024 (Online) 1st February Evening Shift

MCQ (Single Correct Answer)

-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 :

-1

JEE Main 2024 (Online) 1st February Morning Shift

MCQ (Single Correct Answer)

-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 :

$\frac{\sqrt{2} \pi^2}{8}$

$\frac{\sqrt{2} \pi^2}{16}$

$\frac{\sqrt{2} \pi^2}{32}$

$\frac{\sqrt{2} \pi^2}{64}$

JEE Main 2024 (Online) 31st January Evening Shift

MCQ (Single Correct Answer)

-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 :