Definite Integrals — JEE Advanced PYQs

The value of the definite integral

If

then the value of $\sqrt{7}\tan \left(\frac{2\alpha \sqrt{7}}{\pi }\right)$ is ____.
(Here, the inverse trigonometric function ${\tan }^{-1}x$ assumes values in …

Let the function $f:[1,\infty )\to \mathbb{R}$ be defined by …

Let $f:\left[0,\frac{\pi }{2}\right]\to [0,1]$ be the function defined by $f(x)={\sin }^{2}x$ and let $g:\left[0,\frac{\pi }{2}\right]\to [0,\infty )$ be the function defined by $g(x)=\sqrt{\frac{\pi x}{2}-x^2}$.

The value of …

Let $f:\left[0,\frac{\pi }{2}\right]\to [0,1]$ be the function defined by $f(x)={\sin }^{2}x$ and let $g:\left[0,\frac{\pi }{2}\right]\to [0,\infty )$ be the function defined by $g(x)=\sqrt{\frac{\pi x}{2}-x^2}$.

The value of …

For $\mathrm{x}\in \mathbb{R}$, let ${\tan }^{-1}(\mathrm{x})\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then the minimum value of the function $\mathrm{f}:\mathbb{R}\to \mathbb{R}$ defined by …

Consider the equation

Which of the following statements is/are TRUE ?

The greatest integer less than or equal to

is ____.

Let ${\mathrm{g}}_i:\left[\frac{\pi }{8},\frac{3\pi }{8}\right]\to \mathbb{R},\mathrm{i}=1,2$, and $f:\left[\frac{\pi }{8},\frac{3\pi }{8}\right]\to \mathbb{R}$ be functions such that …

Let ${\mathrm{g}}_i:\left[\frac{\pi }{8},\frac{3\pi }{8}\right]\to \mathbb{R},\mathrm{i}=1,2$, and $f:\left[\frac{\pi }{8},\frac{3\pi }{8}\right]\to \mathbb{R}$ be functions such that …

For any real number x, let [x] denote the largest integer less than or equal to x. If

then the value of 9I is ____.

Which of the following inequalities is/are TRUE?

Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function such that its derivative $f^{\prime }$ is continuous and $f(\pi )=-6$. If $F:[0,\pi ]\to \mathbb{R}$ is defined by $F(x)={\int }_0^{x}f(t)dt$, and if …

If

then $27{\mathrm{I}}^2$ equals ____

The value of the integral ${\int }_0^{\pi /2}\frac{3\sqrt{\cos \theta }}{(\sqrt{\cos \theta }+\sqrt{\sin \theta }{)}^5}\mathrm{d}\theta$ equals

The value of the integral

is ____.

Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function such that $f(0)=0,f\left(\frac{\pi }{2}\right)=3$ and $f^{\prime }(0)=1$. If

for …

If $g(x)={\int }_{\sin x}^{\sin (2x)}{\sin }^{-1}(t)dt$, then

If $I={\sum }_{k=1}^{98}{\int }_k^{k+1}\frac{k+1}{x(x+1)}dx$, then

The value of ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\frac{x^2\cos x}{1+e^x}dx$ is equal to

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous odd function, which vanishes exactly at one point and $f(1)=\frac{1}{2}$. Suppose that $F(x)={\int }_{-1}^{x}f(t)dt$ for all $x\in [-1,2]$ and $G(x)={\int }_{-1}^{x}t|f(f(t))|dt$ for all $x\in [-1,2]$. If …

The option(s) with the values of $a$ and $L$ that satisfy the following equation is(are)

Let $f(x)=7{\tan }^{8}x+7{\tan }^{6}x-3{\tan }^{4}x-3{\tan }^{2}x$ for all $x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then the correct expression(s) is(are)

Let $f^{\prime }(x)=\frac{192x^3}{2+{\sin }^4\pi x}$ for all $x\in \mathbb{R}$ with $f\left(\frac{1}{2}\right)=0$. If $m\le {\int }_{1/2}^{1}f(x)dx\le M$, then the possible values of $m$ and $M$ are

PARAGRAPH 2
Let $F:\mathbb{R}\to \mathbb{R}$ be a thrice differentiable function. Suppose that $F(1)=0,F(3)=-4$ and $F^{\prime }(x)<0$ for all $x\in (1/2,3)$. Let $f(x)=xF(x)$ for all $x\in \mathbb{R}$.

If …

Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined by $f(x)=\{\begin{array}{ll}[x],& x\le 2\\ 0,& x>2\end{array}$,
where $[x]$ is the greatest integer less than or equal to $x$. If $I={\int }_{-1}^2\frac{xf\left(x^2\right)}{2+f(x+1)}dx$, then …

Let $F(x)={\int }_x^{x^2+\frac{\pi }{6}}2{\cos }^{2}tdt$ for all $x\in \mathbb{R}$ and $f:\left[0,\frac{1}{2}\right]\to [0,\infty )$ be a continuous function. For $a\in \left[0,\frac{1}{2}\right]$, if $F^{\prime }(a)+2$ is the area of the region bounded by …

Let $f:(0,\infty )\to \mathbb{R}$ be given by

Then

The following integral

is equal to

Let $f:[0,2]\to \mathbb{R}$ be a function which is continuous on $[0,2]$ and is differentiable on ( 0,2 ) with $f(0)=1$. Let

for $x\in [0,2]$. If $F^{\prime }(x)=f^{\prime }(x)$ for all $x\in (0,2)$, then $F(2)$ equ …

Given that for each $a\in (0,1)$,

exists. Let this limit be $g(a)$. In addition, it is given that the function $g(a)$ is differentiable on (0,1).

The value of $g\left(\frac{1}{2}\right)$ is

Given that for each $a\in (0,1)$,

exists. Let this limit be $g(a)$. In addition, it is given that the function $g(a)$ is differentiable on (0,1).

The value of $g^{\prime }\left(\frac{1}{2}\right)$ is

| Q. | The number of points in the interval …

The value of the integral ${\int }_{-\pi /2}^{\pi /2}\left(x^2+\ln \frac{\pi +x}{\pi -x}\right)\cos x\mathrm{d}x$ is

The value of ${\int }_{\sqrt{\ln 2}}^{\sqrt{\ln 3}}\frac{x\sin x^2}{\sin x^2+\sin \left(\ln 6-x^2\right)}dx$ is

Let $f:[-1,2]\to [0,\infty )$ be a continuous function such that $f(x)=f(1-x)$ for all $x\in [-1,2]$. Let $R_1={\int }_{-1}^{2}xf(x)dx$, and $R_2$ be the area of the region bounded by $y=f(x),x=-1,x=2$, and the $x$-axis. Then

The value(s) of ${\int }_0^1\frac{x^4(1-x{)}^4}{1+x^2}dx$ is (are)

For any real number x, let [x] denote the largest integer less than or equal to x. Let f be a real valued function defined on the interval [-10, 10] by

Then the value …

If

then

Let

for $n=1,2,3,\cdots$. Then,

Consider the functions defined implicitly by the equation $y^3-3y+x=0$ on various intervals in the real line.
If $x\in (-\infty ,-2)\cup (2,\infty )$, the equation implicitly defines a unique real valued differentiable function $y=f(x)$.
If $x\in (-2,2)$, the equation implicitl …

Consider the function $f:(-\infty ,\infty )\to (-\infty ,\infty )$ defined by


Let

Which of the following is true?

Match the integrals in Column I with the values in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.
| (A) ${\int }_{-1}^1\frac{dx}{1+x^2}$ | (p) …

The value of $5050\frac{{\int }_0^1{\left(1-x^{50}\right)}^{100}dx}{{\int }_0^1{\left(1-x^{50}\right)}^{101}dx}$ is