Methods of Integration

$\int \frac{x+\sqrt{1-x^2}}{x\sqrt{1-x^2}}dx= ?$

Find $\int \frac{1}{4{\sin }^{2}x+9{\cos }^{2}x}dx$

Evaluate: $\int \frac{\sqrt{x}}{{\left(1+x\right)}^2}dx$, $x>0$

Evaluate the integral $\int \left(2e^{-x/3}\right)dx$.

Evaluate $\int \frac{dx}{x^{5/6}\left(1+5x^{1/3}\right)\sqrt{1+4x^{1/3}}}$

Find the integration of $\frac{1}{\sqrt{x^2-a^2}}$ w.r.t $x$ and hence evaluate $\int \frac{1}{\sqrt{x^2+6x-7}}dx$

Integrate $\int \sqrt{\tan x}dx$

Find $\int \frac{1}{4{\sin }^{2}x+9{\cos }^{2}x}dx$

Evaluate: ${\int }_{1/\sqrt{2}}^1\frac{\left(e^{{\cos }^{-1}x}\right)\left({\sin }^{-1}x\right)}{\sqrt{1-x^2}}dx$

Use the following integrals to match the integral to the correct trig substitution.

Finf the value of $\underset{n\to \infty }{\lim }\left[\frac{1}{n}+\frac{\sqrt{n^2-1^2}}{n^2}+\frac{{\displaystyle \sqrt{n^2-2^2}}}{{\displaystyle n^2}}+......+\frac{{\displaystyle \sqrt{n^2-{\left(n-1\right)}^2}}}{{\displaystyle n^2}}\right]$

Evaluate the following $\mathrm{I}=\int \frac{{\tan }^2{\mathrm{xsec}}^2\mathrm{x}}{1-{\tan }^6\mathrm{x}}\mathrm{dx}$.

Evaluate $\int {\cot }^{-1}\left(\frac{1+x-x^2}{2x-1}\right)dx$

Evaluate: $\int \frac{1}{x\sqrt{ax-x^2}}dx$

Using integration, prove that:

$\int \frac{1}{\sqrt{x^2-a^2}}dx=\log \left|x+\sqrt{x^2-a^2}\right|+C$

Prove that $\underset{n\to \infty }{\lim }\left[\frac{1}{n}+\frac{\sqrt{n^2-1^2}}{n^2}+\frac{{\displaystyle \sqrt{n^2-2^2}}}{{\displaystyle n^2}}+......+\frac{{\displaystyle \sqrt{n^2-{\left(n-1\right)}^2}}}{{\displaystyle n^2}}\right]=\frac{\mathrm{\pi }}{4}$

If $\int x^5\cos \left(x^6\right)dx=k\sin \left(x^6\right)+c$, find the value of $k$.

$\int \sin \mathrm{x}\cdot \cos \mathrm{x}\cdot \cos 4\mathrm{x}\cdot \cos 8\mathrm{x}\cdot \cos 16\mathrm{x} \mathrm{dx}$

Find $\int \frac{{\left(x^4-x\right)}^{\frac{1}{4}}}{x^5}dx$

Find the following integrals: $\int \frac{x+3}{\sqrt{5-4x-x^2}}dx$