Indefinite Integration

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

For the following two (02) items: Let $u = \int e^x \cos x\, dx$ and $v = \int e^x \sin x\, dx$. Consider the following: I. $\dfrac{du}{dx} = -v$ II. $\dfrac{dv}{dx} = -u$ Which of the above is/are correct?

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 $A$ ?

What is $\int \frac{(\cos x)^{1.5}-(\sin x)^{1.5}}{\sqrt{\sin x \cdot \cos x}} d x$ equal to?

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 $B$ ?

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 $x^2 ?$

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

If $I_1=\int \frac{e^x d x}{e^x+e^{-x}}$ and $I_2=\int \frac{d x}{e^{2 x}+1}$, then what is $I_1+I_2$ equal to?

What is $\int \frac{d x}{a^2 \sin ^2 \mathrm{x}+\mathrm{b}^2 \cos ^2 \mathrm{x}}$ equal to?

What is $\int \tan ^{-1}(\sec x+\tan x) d x$ equal to?