Definite integration practice · Medium

If the orthogonal trajectories of the family of curves $e^{n-1} x=y^n$ always passes through $(1, e)$ and $(0,0)$, then value of $\boldsymbol{n}$ is

Area inside the parabola $y^2=4 a x$, between the lines $x=a$ and $x=4 a$ is equal to

If $\left[ x \right]$ is the greatest integer $\le x$ then ${\pi ^2}\int\limits_0^2 {\left( {\sin \frac{{\pi x}}{2}} \right)(x - \left| x \right|} {)^{\left[ x \right]}}dx$ is equal to

The value of the integral $\int\limits_{0}^1 \frac{x^\alpha-1}{\ln x} d x$, is

If $\int \limits_0^1 \,\,{\cot ^{ - 1}}\,\,\left( {1\,\, - \,\,x\,\, + \,\,{x^2}} \right)\,\,dx = k\,\int \limits_0^1 {\tan ^{ - 1}}\,\,x\,dx$, then $k =$

$\int \nolimits_{ - \pi /2}^{\pi /2} \,\,\sin \,\,\left| {\,x\,} \right|\,\,dx$is equal to

Find the value of $\int\limits_{{-1}}^{3 / 2}|x \sin \pi x| d x$.

The value of $\int \limits_3^5 \,\,\frac{{{x^2}}}{{{x^2} - \,\,4}}\,\,dx$, is

The value of $\int \limits_0^{\pi /2} \,\mathrm{cosec}\;\,\,\left( {x - \pi /3} \right)\,\,\mathrm{cosec}\,\,\left( {x\,\, - \,\,\pi /6} \right)dx$, is

The value of the integral $\int\limits_{0}^x(1-|\sin 8 x|) d x$ is

If $y=f(x)$ makes positive intercepts of 2 and 1 unit on $x$ and $y$ coordinate axes and encloses an area of $3 / 4$ square unit with the axes, then $\int\limits_{0}^2 x f^{\prime}(x) d x$, is

The value of $\int \limits_1^{\sqrt[7]{2}} \,\,\frac{1}{{x\,\,\left( {2{x^7}\,\, + \,\,1} \right)}}\,\,dx$, is

The value of integral $\int\limits_ {\frac{3 \pi}{4}}^{\frac{3 \pi}{4}} \frac{x}{1+\sin x} d x$ is

Let $f(x) =\frac{e^x}{1+e^x}$ $I_1 =\int\limits_{{f(-a)}}^{f(a)} x g(x(1-x)) d x$ $I_2 =\int\limits_{{f(-a)}}^{f(a)} g(x(1-x)) d x$ then $I_2 / I_1$ is

If $f\left( x \right)\mathrm{\;}\,\,\mathrm{and}\,\,\;g\left( x \right),\,\,\;x\, \in \,R$ are continuous functions, then value of integral

$\int \nolimits_{ - \pi /2}^{\pi /2} \,\,\left[ {\left\{ {\,f\left( x \right)\,\, + \,\,f\left( { - x} \right)} \right\}\,\,\left\{ {g\left( x \right)\,\, - \,\,g\left( { - x} \right)} \right\}} \right]\;dx$ is

If $F\left( x \right) = \frac{1}{{{x^2}}}\,\,\int \limits_4^x \,\,\left[ {4{t^2}\,\, - \,\,2\,F'\,\,\left( t \right)} \right]\,dt$, then $F'\,\,\left( 4 \right)$ equals

Let $I_n=\int_0^{\pi / 2} \frac{\sin (2 n-1) x}{\sin x} d x, \mathcal{F}_n=\int_0^{\pi / 2} \frac{\sin ^2 n x}{\sin ^2 x}, n \in N$, then

The value of integral $\int \nolimits_0^1 \,\,\sqrt {\frac{{1\,\, - \,\,x}}{{1\,\, + \,\,x}}} \;\,\,dx$ is

Let $n$ be the number of integral values of $x$, which satisfy the equation

$\tan ^{-1} x-\sin ^{-1} x=\cos ^{-1} x-\cot ^{-1} x$.

Let $\vec{c}=|\vec{a} \times \vec{b}| \vec{a}+(\vec{a} \cdot \vec{b}) \vec{b} \quad$ where $\vec{a}$ and $\vec{b}$ are unit vectors and angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{6}$

Let the value of definite integral $\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{\ln (\sec x)}{(2013)^{\sin x}+1} d x$ equals $\frac{\pi}{2} \ln k$, then the value of $\left(\mathrm{n}+2|\overrightarrow{\mathrm{c}}|^2+\mathrm{k}\right)$ is equal to

If $f(x)=\left\{\begin{array}{l}e^{\cos x} \sin x \text { for }|x| \leq 2 \\ 2 \text { otherwise}\end{array}\right.$, then $\int\limits_{{-2}}^3 f(x) d x=$

The area bounded by the curve $y=4 x-x^2$ and the $x$-axis, is

The value of the definite integral $\int\limits_{0}^1 \frac{x d x}{\left(x^3+16\right)}$ lies in the interval $[a, b]$. Then smallest such interval is

$\frac{3(e-1)}{e} \int\limits_{1}^2 x^2 e^{[x]+\left[x^2\right]} dx$ equals:

$\int\limits_{0}^5[x]\{x\} d x$ (where [.] denotes greatest integer function and $\{$.$\}$ denotes fractional part of function) is

If $\phi \left( x \right) = \cos \,\,\,x\, - \int \nolimits_0^x \,\,\left( {x - t} \right)\,\,\phi \,\,\left( t \right)\,\,dt$, then $\phi ''\,\,\left( x \right)\,\, + \,\,\phi \,\,\left( x \right)$ is equal to

If $\int \frac{\sin x}{\sin (x-\alpha)} d x=A x+B \log \sin (x-\alpha),+C$, then value of $(A, B)$ is