JEE Integration PYQs with Solutions PDF, Download Now

Let $$f(x)=\int\frac{(2-x^{2}).e^{x}}{(\sqrt{1+x})(1-x)^{\frac{3}{2}}}dx$$. If f(0)=0, then $$f\left(\frac{1}{2}\right)$$ is equal to:

If $$\int (\sin x) ^{\frac{-11}{2}}(\cos x)^{\frac{-5}{2}}dx= -\frac{p_{1}}{q_{1}}(\cot x)^{\frac{9}{2}}-\frac{p_{2}}{q_{2}}(\cot x)^{\frac{5}{2}}-\frac{p_{3}}{q_{3}}(\cot x)^{\frac{1}{2}}+ \frac{p_{4}}{q_{4}}(\cot x)^{\frac{-3}{2}}+C,\text{ where }p_{i} \text{ and } q_{i} $$ are positive integers with $$gcd(p_{i}, q_{i}) = l$$ for i = l, 2, 3, 4 and C is the constant of integration, then $$\frac{15p_{1}p_{2}p_{3}p_{4}}{q_{1}q_{2}q_{3}q_{4}} $$ is equal to ______

Let $$f(t)=\int_{}^{}\left(\frac{1-\sin(\log_{e}{t})}{1-\cos(\log_{e}{t})}\right)dt,t > 1$$.
If $$f(e^{\pi/2})=-e^{\pi/2}\text{ and }f(e^{\pi/4})=\alpha e^{\pi/4}$$, then $$\alpha$$ equals

If $$\int_{}^{}\left(\frac{1-5\cos^{2} x}{\sin^{5} x \cos^{2} x}\right)dx=f(x)+C$$, where C is the constant of integration, then $$f\left(\frac{\pi}{6}\right)-f\left(\frac{\pi}{4}\right)$$ is equal to

Let $$I(x)=\int\frac{3dx}{\left(4x+6\right)\left(\sqrt{4x^{2}}+8x+3\right)}$$ and $$I(0)=\frac{{\sqrt{3}}}{4}+20.$$
If $$I\left( \frac{1}{2} \right)=\frac{a\sqrt{2}}{b}+c, \text { Where a,b,c } \in N,gcd(a,b)=1, \text{ a+b+c is equal to}$$

Let $$f(x)=\int\frac{dx}{x^{\left(\frac{2}{3}\right)}+2x^{\left(\frac{1}{2}\right)}} $$ be such that $$f(0)=-26+24\log_{e}{(2)}. \text { If } f(1)=a+b \log_{e}{(3)}, \text{ where } a,b \in Z$$, then a+b is equal to:

If $$\int_{}^{}e^{x}\left(\frac{x\sin^{-1}x}{\sqrt{1-x^{2}}}+\frac{sin^{-1}x}{(1-x^{2})^{3/2}}+\frac{x}{1-x^{2}}\right)dx=g(x)+C$$ where C is the constant of integration, then $$g(\frac{1}{2})$$ equals

Let $$\int_{}^{} x^{3}\sin x dx =g(x)+C$$, where is the constant of integration. If $$8(g(\frac{\pi}{2})+g'(\frac{\pi}{2}))=\alpha \pi^{3} + \beta \pi^{2} + \gamma ,\alpha ,\beta ,\gamma \in Z$$, then $$\alpha +\beta - \gamma$$ equals :

If $$\int \frac{2x^2+5x+9}{\sqrt{x^2+x+1}}\,dx=x\sqrt{x^2+x+1}+\alpha\sqrt{x^2+x+1}+\beta\log_e\!\left|x+\frac12+\sqrt{x^2+x+1}\right|+C$$, where $$C$$ is the constant of integration, then $$\alpha+2\beta$$ is equal to $$\underline{\hspace{2cm}}.$$

Let $$I(x)=\int_{}^{} \frac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$$. If $$I(37)-I(24)=\frac{1}{4}\left(\frac{1}{b^{\frac{1}{13}}}-\frac{1}{c^{\frac{1}{13}}}\right),b,c\in N$$, then 3(b+c) is equal to

If $$f(x)=\int_{}^{}\frac{1}{x^{1/4}(1+x^{1/4})}dx,f(0)=-6$$, then $$f(1)$$ is equal to :

The integral $$\int \frac{x^8 - x^2}{(x^{12} + 3x^6 + 1)\tan^{-1}\left(x^3 + \frac{1}{x^3}\right)} dx$$ is equal to :

For $$x \in (-\frac{\pi}{2}, \frac{\pi}{2})$$, if $$y(x) = \int \frac{\csc x + \sin x}{\csc x \sec x + \tan x \sin^2 x} dx$$ and $$\lim_{x \to (\frac{\pi}{2})^-} y(x) = 0$$ then $$y(\frac{\pi}{4})$$ is equal to

If $$\int \frac{\sin^{\frac{3}{2}}x + \cos^{\frac{3}{2}}x}{\sqrt{\sin^3 x \cos^3 x \sin(x - \theta)}} dx = A\sqrt{\cos\theta\tan x - \sin\theta} + B\sqrt{\cos\theta - \sin\theta\cot x} + C$$, where $$C$$ is the integration constant, then $$AB$$ is equal to

If $$\int \csc^5 x \, dx = \alpha \cot x \csc x \left(\csc^2 x + \frac{3}{2}\right) + \beta \log_e \left|\tan \frac{x}{2}\right| + C$$ where $$\alpha, \beta \in \mathbb{R}$$ and $$C$$ is the constant of integration, then the value of $$8(\alpha + \beta)$$ equals _____

If $$\int \frac{1}{a^2 \sin^2 x + b^2 \cos^2 x} dx = \frac{1}{12} \tan^{-1}(3 \tan x) +$$ constant, then the maximum value of $$a \sin x + b \cos x$$, is :

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

If $$\int \frac{1}{\sqrt[5]{(x-1)^4(x+3)^6}} dx = A\left(\frac{\alpha x - 1}{\beta x + 3}\right)^B + C$$, where C is the constant of integration, then the value of $$\alpha + \beta + 20AB$$ is _____

Let $$\int \frac{2 - \tan x}{3 + \tan x} dx = \frac{1}{2}(\alpha x + \log_e|\beta \sin x + \gamma \cos x|) + C$$, where $$C$$ is the constant of integration. Then $$\alpha + \frac{\gamma}{\beta}$$ is equal to :

If the area of the region $$\left\{\left(x,y\right): 1-2x \leq y \leq4-x^{2}, x\geq 0, y\geq0 \right\}$$ is $$\frac{\alpha}{\beta} , \alpha,\beta \epsilon N$$, gcd $$\left(\alpha,\beta\right)=1$$, then the value of $$\left(\alpha+\beta\right)$$ is

If $$\displaystyle \int_{0}^{1} 4\cot^{-1}(1 - 2x + 4x^2)\,dx = a\tan^{-1}(2) - b\log_e(5)$$, where $$a,b\epsilon N$$ then (2a + b} is equal to _________

Let [·] denote the greatest integer function and $$f(x) = \lim_{n \to \infty} \frac{1}{n^3} \sum_{k=1}^{n} \left[\frac{k^2}{3^x}\right]$$. Then $$12 \sum_{j=1}^{\infty} f(j)$$ is equal to _______

The area of the region, inside the ellipse $$x^{2}+4y^{2}=4$$ and outside the region bounded by the curves y=|x|-1 and y=1-|x|, is:

The value of $$\int_{\frac{-\pi}{6}}^{\frac{\pi}{6}}\left(\frac{\pi+4x^{11}}{1-\sin(|x|+\frac{\pi}{6})}\right)dx$$ is equal to :

$$6\int_{0}^{\pi} |(\sin3x+\sin2x+\sin x)|dx$$ is equal to _________.

The area of the region $$A= \left\{(x,y): 4x^{2}+y^{2}\leq 8 \text{and } y^{2} \leq 4x \right\}$$ is :

Let $$\left[\cdot\right]$$ be the greatest integer function. If $$(\alpha = \int_{0}^{64} \left( x^{1/3} - [x^{1/3}] \right)\, dx $$, then $$\frac{1}{\pi} \int_{0}^{\alpha\pi } \left( \frac{\sin^{2}\theta } {\sin^{6}\theta + \cos^{6}\theta} \right) d\theta$$ is equal to ____ .

The value of the integral $$\int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1+\sqrt[3]{\tan2x}}$$ is:

Let f be a twice differentiable non-negative function such that $$(f(x))^{2}=25+\int_{0}^{x}\left((f(t))^{2}+(f'(t))^{2}\right)dt$$. Then the mean of $$f(\log_{e}{(1)}),f(\log_{e}{(2)}),.....,f(\log_{e}{(625)})$$ is equal to:

Let the area of the region bounded by the curve y= max $${\sin x, \cos x}$$, lines x = O, $$x=\frac{3\pi}{2}$$, and the x-axis be A. Then, A+$$A^{2}$$ is equal to_____.

Let $$ f: [1 , \infty ) \rightarrow R$$ be a differentiable function. If $$6 \int_{1}^{x} f(t)dt=3x f(x)+ x^{3}-4$$ for all $$x\geq 1$$ then the value of $$f(2)-f(3)$$ is

The value of $$ \int_{\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \frac{1}{[x]+4}\right)dx $$ where [ . ]denotes the greatest integer function, is

Let the line $$x = - 1$$ divide the area of the region $$ \left\{(x,y): 1+x^{2}\leq y \leq 3 -x\right\} $$ in the ratio m : n, gcd (m, n) = 1. Then m + n is equal to

Let a differentiable function f satisfy the equation $$\int_{0}^{36}f(\frac{tx}{36})dt=4\alpha f(x)$$. If y=f(x) is a standard parabola passing through the points (2, 1) and $$(-4,\beta)$$, then $$/beta^{\alpha}$$ is equal to______.

Let f(a) denote the area of the region in the first quadrant bounded by x = 0, x = 1, $$y^{2}=x$$ and y = |ax - 5| - |1 - ax| + $$ax^{2}$$. Then (f(O) + f(1)) is equal

Let f be a polynomial function such that $$f(x^{2}+1)=x^{4}+5x^{2}+2$$, for all $$x \in R$$. Then $$\int_{0}^{3}f(x)dx$$ is equal to

The value of $$\sum_{r=1}^{20}\left(\left|\sqrt{\pi\left(\int_{0}^{r}x|\sin \pi x\right)}\right|\right)$$ is ______.

The area of the region enclosed between the circles $$x^{2}+y^{2}=4 \text{ and } x^{2}+(y-2)^{2}=4$$ is

The number of elements in the
set $$ S=\left\{ x:x\in [0,100] \text{ and } \int_{0}^{x} t^{2} \sin(x-t)dt=x^{2}\right\}$$ is _________

Let $$[\cdot]$$ denote the greatest integer function. Then $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(\frac{12(3+[x])}{3+[\sin x]+[\cos x]}\right)dx$$ is equal to:

Let $$ f$$ be a differentiable function satisfying $$f(x)=1-2x+\int_{0}^{x} e^{(x-t)}f(t)dt, x \in \mathbb{R}$$ and let $$g(x)=\int_{0}^{x} (f(t)+2)^{15}(t-4)^{6}(t+12)^{17}dt, x \in \mathbb{R}.$$ If p and q are respectively the points of local minima and local maxima of g, then the value of $$\mid p+q \mid $$ is equal to _________

The area of the region enclosed by the curves $$y=x^{2}-4x+4\text{ and }y^{2}=16-8x$$ is :

$$ \text{Let for }f(x)=7 \tan^{8}x + 7\tan^{6}x-3\tan^{4}x-3\tan^{2}x \text{ } I_1=\int_{0}^{\pi/4}f(x)dx \text{ and }I_2=\int_{0}^{\pi/4}xf(x)dx. \text{ Then } 7I_1+12T_2 \text{ is equal to :} $$

$$ \text{The area of the region, inside the circle }(x-2\sqrt{3})^{2}+y^{2}=12 \text{ and outside the parabola } y^{2}=2\sqrt{3}x \text{ is :} $$

Let the function, $$f(x)=\begin{cases}-3ax^{2}-2, & x < 1\\a^{2}+bx, & x \geq 0\end{cases}$$ be differentiable for all  $$x \in R,  $$ where $$ a>1, b \in R$$.  If the area of the region enclosed by $$ y=f(x) \text{and the line } y= -20 \text{ is } \alpha+\beta\sqrt{3},\alpha, \beta \in Z$$,  then the value of  $$\alpha + \beta \text{ is } $$_______

If $$I=\int_{0}^{\frac{\pi}{2}}\frac{\sin^{\frac{3}{2}} x}{\sin^{\frac{3}{2}} x+ \cos^{\frac{3}{2} x}}dx$$, then $$\int_{0}^{21}\frac{x\sin x \cos x}{\sin^{4} x+\cos^{4} x}dx$$ equals :

If the area of the region $$\{(x,y):-1 \leq x \leq 1, 0 \leq y \leq a + e^{|x|}-e^{-x}, a > 0\}$$ is $$\frac{e^{2}+8e+1}{e}$$, then the value of a is :

The area of the region $$\left\{(x,y) : x^2 + 4x + 2 \le y \le |x+2| \right\}$$ is equal to:

$$ \text{If } I(m,n)=\int_{0}^{1} x^{m-1}(1-x)^{\,n-1}\,dx,\quad m,n>0, \text{ then } I(9,14)+I(10,13) \text{ is:} $$

Let $$f$$ be a differentiable function such that $$2(x+2)^2f(x)-3(x+2)^2 = 10\int_0^x (t+2)f(t)\,dt,\quad x\ge0.$$ Then $$f(2)$$ is equal to: $$\underline{\hspace{1cm}}$$