JEE MAIN Definite Integration Previous Year Questions (PYQs) – Page 9 of 17

$\displaystyle \int_{\tfrac{3\sqrt{2}}{4}}^{\tfrac{3\sqrt{3}}{4}} \dfrac{48}{\sqrt{9-4x^{2}}}\,dx$ is equal to:

Let $I(x)=\displaystyle\int \frac{6}{\sin^{2}x,(1-\cot x)^{2}},dx$. If $I(0)=3$, then $I!\left(\tfrac{\pi}{12}\right)$ is equal to

Which of the following statements is correct for the function g($\alpha$) for $\alpha$ $\in$ R such that $g(\alpha ) = \int\limits_{{\pi \over 6}}^{{\pi \over 3}} {{{{{\sin }^\alpha }x} \over {{{\cos }^\alpha }x + {{\sin }^\alpha }x}}dx} $

Let the function $f:[0,2]\to\mathbb{R}$ be defined as \[ f(x)= \begin{cases} e^{\min\{x^2,\; x-[x]\}}, & x\in[0,1),\\[4pt] e^{[\,x-\log_e x\,]}, & x\in[1,2], \end{cases} \] where $[t]$ denotes the greatest integer less than or equal to $t$. Then the value of the integral $\displaystyle \int_{0}^{2} x f(x)\,dx$ is:

Let $f:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=\dfrac{x}{(1+2x^{4})^{1/4}}$, and $g(x)=f(f(f(f(x))))$. Then $18\displaystyle\int_{0}^{\sqrt{2\sqrt{5}}} x^{2}g(x),dx$ is equal to:

For $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, if $y(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname{cosec} x \sec x+\tan x \sin ^2 x} d x$, and $\lim _\limits{x \rightarrow\left(\frac{\pi}{2}\right)^{-}} y(x)=0$ then $y\left(\frac{\pi}{4}\right)$ is equal to

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 $\dfrac{\pi}{6}$ and $\dfrac{\pi}{4}$ respectively with the positive $x$-axis. If $27\displaystyle\int_{1}^{3}\big((f'(t))^{2}+1\big)f'''(t),dt=\alpha+\beta\sqrt{3}$, where $\alpha,\beta$ are integers, then the value of $\alpha+\beta$ equals:

The area of the region in the first quadrant inside the circle $x^{2}+y^{2}=8$ and outside the parabola $y^{2}=2x$ is equal to

Let f be a differentiable function in $\left( {0,{\pi \over 2}} \right)$. If $\int\limits_{\cos x}^1 {{t^2}\,f(t)dt = {{\sin }^3}x + \cos x} $, then ${1 \over {\sqrt 3 }}f'\left( {{1 \over {\sqrt 3 }}} \right)$ is equal to

The integral $\int\limits_0^1 {{1 \over {{7^{\left[ {{1 \over x}} \right]}}}}dx} $, where [ . ] denotes the greatest integer function, is equal to