Indefinite Integration

The integral $\int\,\frac{dx}{x^{2}\left(x^{4}+1\right)^{\frac{3}{4}}}$equals to

If $\int\,\frac{dx}{\left(x^{2}+x+1\right)^{2}}=a\tan\,^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)+b\left(\frac{2x+1}{x^{2}+x+1}\right)+C,x>0$where $C$ is theconstant of integration, then the value of $9(\sqrt{3}a+b)$ is…

Let $f(t)=\int\left(\frac{1-\sin \left(\log _{e} t\right)}{1-\cos \left(\log _{e} t\right)}\right) d t, t>1$. If $f\left(e^{\pi / 2}\right)=-e^{\pi / 2}$ and $f\left(e^{\pi / 4}\right)=\alpha e^{\pi / 4}$, then $\alpha$…

If $\int(\sin x)^{\frac{-11}{2}}(\cos x)^{\frac{-5}{2}} d x=$ $-\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…

If $\int\,\frac{\sin\,x}{\sin\,^{3}x+\cos\,^{3}x}dx=\alpha\,\log\,_{e}|1+\tan\,x|+\beta\,\log\,_{e}\left|1-\tan\,x+\tan\,^{2}x\right|+\gamma\,\tan\,^{-1}\left(\frac{2\tan\,x-1}{\sqrt{3}}\right)+C,$when $C$ is constant…

If $\int\,\sin\,^{-1}\left(\frac{\sqrt{x}}{1+x}\right)dx=A\left(x\right)\tan\,^{-1}\left(\sqrt{x}\right)+B\left(x\right)+C$, where$C$is a constant of integration, then the orderedpair…

For real numbers $\alpha\,,\beta\,,\gamma\,$ and $\delta\,,$…

If $\int \operatorname{cosec}^5 x d x=\alpha \cot x \operatorname{cosec} x\left(\operatorname{cossc}^2 x+\frac{3}{2}\right)+\beta \log _\epsilon\left|\tan \frac{x}{2}\right|+C$ where $\alpha, \beta \in \mathbb{R}$ and…

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

What is $\int \frac{d x}{\sec ^2\left(\tan ^{-1} x\right)}$ equal to ?

What is the integral of $f(x)=1+x^2+x^4$ with respect to $x^2 ?$

Passage: Let $\displaystyle\int\dfrac{\sin\theta\,d\theta}{(2+\cos\theta)(3+4\cos\theta)}=A\ln|2+\cos\theta|+B\ln|3+4\cos\theta|$. Question: What is the value of $A$ ?