List of top Mathematics Questions on Definite Integral asked in JEE Advanced

For $x ∈ R$, let $tan^{-1} (x) ∈$ ($-\frac{\pi}{2},\frac{\pi}{2}$). Then the minimum value of function $f: R \rightarrow R$ defined by $f(x) =$ $\int_{0}^{x\,tan^{-1}x}\frac{e^{(t-cos\,t)}}{1+t^{2023}}dt$  is

Consider the equation $\int\limits_1^e \frac{\left(\log _{ e } x \right)^{1 / 2}}{x\left(a-\left(\log _{ e } x \right)^{3 / 2}\right)^2} dx =1, \quad a \in(-\infty, 0) \cup(1, \infty)$ Which of the following statements is/are TRUE ?

The greatest integer less than or equal to $\int\limits_1^2 \log _2\left(x^3+1\right) d x+\int\limits_1^{\log _2 9}\left(2^x-1\right)^{\frac{1}{3}} dx$ is _____.

The greatest integer less than or equal to $\int\limits_1^2 \log _2\left(x^3+1\right) d x+\int\limits_1^{\log _2 9}\left(2^x-1\right)^{\frac{1}{3}} dx$ is _____.

Consider the equation $\int\limits_1^e \frac{\left(\log _{ e } x \right)^{1 / 2}}{x\left(a-\left(\log _{ e } x \right)^{3 / 2}\right)^2} dx =1, \quad a \in(-\infty, 0) \cup(1, \infty)$ Which of the following statements is/are TRUE ?

For any real number $x$, let $[x]$ denote the largest integer less than or equal to $x$ If
$I=\int\limits_{0}^{10}\left[\sqrt{\frac{10 x}{x+1}}\right] d x,$
then the value of $9I$ is _____

Let $f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow R$ be a continuous function such that
$f(0)=1$ and $\int\limits_{0}^{\frac{\pi}{3}} f(t) d t=0$
Then which of the following statements is(are) TRUE?

Let $f_{1}:(0, \infty) \rightarrow R$ and $f_{2}:(0, \infty) \rightarrow R$ be defined by
$f_{1}(x)=\int\limits_{0}^{x} \displaystyle\prod_{j=1}^{21}(t-j)^{j} d t,$ x$>$0 
and $f _{2}( x )=98( x -1)^{50}-600( x -1)^{49}+2450, x >0$, 
where, for any positive integer $n$ and real numbers $a _{1}, a _{2}$, $\ldots , a_{n}, \displaystyle\prod_{i=1}^{n} a_{i}$ denotes the product of $a_{1}, a_{2}, \ldots , a_{n}$. Let $m_{i}$ and $n_{i}$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f _{ i }, i =1,2$, in the interval $(0$, $\infty$ )

Let $g _{ i }:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow R , i =1,2$ and $f :\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow R$ be functions such that
$g_{1}(x)=1, g_{2}(x)=|4 x-\pi|$ and $f(x)=\sin ^{2} x$, for all $x \in\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right]$ ,
Define $S_{i}=\int\limits_{\frac{\pi}{8}}^{\frac{3 \pi}{8}} f(x) \cdot g_{i}(x) d x, i=1,2$

Let the function $f: R \rightarrow R$ and $g: R \rightarrow R$ be defined by
$f(x)=e^{x-1}-e^{-|x-1|}$ and $g(x)=\frac{1}{2}\left(e^{x-1}+e^{1-x}\right)$.
Then the area of the region in the first quadrant bounded by the curves $y=f(x), y=g(x)$ and $x=0$ is

If $f(x)=\int_{x^2}^{x^2+1} e^{-t {^2}}dt$ then f(x) increases in

The value of $\int^{ \pi / 2}_0 \frac{ dx }{ 1 + \tan^3 \, x } \, is \,$

The value of the definite integral $\int _0^1 (1+e^{-x^2}) dx \, is$