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

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

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…

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…

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

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

Let $f(x) = \int \left(\dfrac{16x + 24}{x^2 + 2x - 15}\right) dx$. If $f(4) = 14 \log_e(3)$ and $f(7) = \log_e(2^{\alpha} \cdot 3^{\beta})$, $\alpha, \beta \in \mathbb{N}$, then $\alpha + \beta$ is equal to:

Let $f(x)=\int \frac{\mathrm{d} x}{x^{\left(\frac{2}{3}\right)}+2 x^{\left(\frac{1}{2}\right)}}$ be such that $f(0)=-26+24 \log _{\mathrm{e}}(2)$. If $f(1)=\mathrm{a}+\mathrm{b} \log _{\mathrm{e}}(3)$, where…

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 $\alpha$…

Let $\mathrm{I}(x)=\int \frac{3 d x}{(4 x+6)\left(\sqrt{4 x^{2}+8 x+3}\right)}$ and $\mathrm{I}(0)=\frac{\sqrt{3}}{4}+20$. If $\mathrm{I}\left(\frac{1}{2}\right)=\frac{a \sqrt{2}}{b}+\mathrm{c}$, where $a, b, \mathrm{c}…