Indefinite integration practice · Medium

$\int \sin (101 x) \sin ^{99} x d x$ equals

$\int \,\,\frac{{dx}}{{\sin \,\,\left( {x - a} \right)\,\,\sin \left( {x - b} \right)}}$ is

For $\alpha, \beta, \gamma, \delta \in \mathbb{N}$, if $\int\left(\left(\frac{x}{e}\right)^{2 x}+\left(\frac{e}{x}\right)^{2 x}\right) \log _e x$ $d x=\frac{1}{\alpha}\left(\frac{x}{e}\right)^{\beta x}-\frac{1}{\gamma}\left(\frac{e}{x}\right)^{8 x}+C$, where $e=\sum\limits_{{n=0}}^{\infty} \frac{1}{n !}$ and $C$ is constant of integration, then $\alpha+2 \beta+3 \gamma-4 \delta$ is equal to

$\int \,\,\frac{1}{{\sqrt {{{\sin }^3}\,\,x\,\,\sin \,\,\left( {x + \alpha } \right)} }}\,\,dx$ is equal to

$\int \frac{\sin ^3 x}{\left(\cos ^4 x+3 \cos ^2 x+1\right) \tan ^{-1}(\sec x+\cos x)} d x$ is equal to

$\int e^{\sin \theta}\left(\log \sin \theta+\operatorname{cosec}^2 \theta\right) \cos \theta d \theta$ is equal to

$\int \frac{\sin x+\cos x}{9+16 \sin 2 x} d x$ is equal to

$\int e^{2 x}(-\sin x+2 \cos x) d x=$

$\int \frac{d x}{\sqrt{x+a}+\sqrt{x+b}}=$

If $\frac{d}{{dx}}\,\,f\left( x \right) = g\left( x \right)$ for $a \le x \le b$ then, $\int \limits_a^b \,f\left( x \right)\,\,g\,\,\left( x \right)\,\,dx$ equals

$\int \frac{d x}{(x-3)^2 \sqrt{x^2-6 x+4}}$ is equal to

The value of $\int \frac{1}{\sin \left(x-\frac{\pi}{3}\right) \cos x} d x$, is

$\int \,\,\sqrt {{x^2}\,\, + \,\,{a^2}} \;\,dx$ equals

$\int \frac{d x}{\left(x^2+x+1\right) \sqrt{x^2+x-1}}=$

$\int \frac{d x}{\left(4+3 x^2\right) \sqrt{3-4 x^2}}=$

If $\frac{d}{{dx}}\,\,\{ f\left( x \right) = \frac{1}{{1 + {x^2}}}$, then $\frac{d}{{dx}}\,\,\left\{ {f\left( {{x^3}} \right)} \right\}$ is

If $\int \frac{1}{x \sqrt{1-x^3}} d x=a \log \left|\frac{\sqrt{1-x^3}-1}{\sqrt{1-x^3}+1}\right|+b$ then $a$ is equal to

$\int \frac{{dx}}{{x\,\sqrt {{x^6}\,\, - \,\,16} }}$ is equal to

If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a function satisfying the following:
(i) $f(-x)=-f(x)$
(ii) $f(x+1)=f(x)+1$
(iii) $f\left(\frac{1}{x}\right)=\frac{f(x)}{x^2} \forall x \neq 0$
then $\int e^x f(x) d x$ is equal to

$\int \frac{x^2 \cos x}{(1+\sin x)^2} d x= \frac{-x^2}{1+\sin x}-2 x(\sec x-\tan x)+2 \log |f(x)|+c$ where $f(x)$ is equal to

If $\mathrm{A}=\int_0^{505 \pi}|\cos x| d x$ and $B=\int_{505 \pi}^{1007 \pi}|\sin x| d x$, then $A+B$ is equal to

$\int \frac{d x}{\sqrt[3]{x^{5 / 2}(x+1)^{7 / 2}}}$ is equal to:

The value of $\int \frac{\ln (x / e)}{(\ln x)^2} d x$ is

$\int \frac{d x}{\cos (2 x) \cos (4 x)}$ is equal to

$\int \frac{\sqrt{1+x^2}}{x^4} d x$ is equal to

If $m$ is a non-zero number and $\int \frac{x^{5 m-1}+2 x^{4 m-1}}{\left(x^{2 m}+x^m+1\right)^3} d x=f(x)+c$, then $f(x)$ is

An object falling from rest in air is subject not only to the gravitational force but also to air resistance. Assume that the air resistance is proportional to the velocity with constant of proportionality as $k(>0)$ and acts in a direction opposite to motion $\left(g=9.8 \mathrm{~m} / \mathrm{s}^2\right)$. Then velocity at time $t=1 / k$ is

If $f'\,\,\left( x \right) = x\,\, + \,\,\frac{1}{x}$, then the value of $f\left( x \right)$ is

$\int\left[\log (\log x)+\frac{1}{(\log x)^2}\right] d x=$

$\int \frac{d x}{x\left(x^{2010}+1\right)}$ is equal to