Methods of Integration

The integral $\int \left\{\frac{x+2}{\left(x^2+3x+3\right)\sqrt{x+1}}\right\}dx$ is equal to

(where $C$ is a constant of integration)

Let $f\left(x\right)=\frac{x}{{\left(1+x^n\right)}^{{\displaystyle \frac{1}{n}}}}$ for $n\ge 2$ and $g\left(x\right)=\underset{f \text{occurs} n \text{times}}{\underbrace{\left(fofofof.....of\right)}}\left(x\right)$. Then, $\int x^{n-2}g\left(x\right)dx$ equals (where $K$ is an integration constant)

The value of $\int \frac{1}{{\cos }^{6}x+{\sin }^{6}x}dx$ is equal to

$\int {\left(\ln x\right)}^{-1}dx-\int {\left(\ln x\right)}^{-2}dx$ is equal to

What is $\int \frac{x{e}^{x}dx}{{\left(x+1\right)}^2}$ equal to ?

What is the value of $\int a^x{e}^{x}dx$ ?

What is the value of $\int {\left(x^2+1\right)}^{\frac{5}{2}}xdx$ ?

What is $\int \sqrt{x}\left(e^{\sqrt{x}}\right)dx$ equal to?

What is $\int \log \left(x+1\right)dx$ equal to?

What is $\int {\left(e^x+1\right)}^{-1}dx$ equal to ?

What is $\int \sec x°dx$ equal to?

What is $\int {\tan }^{2}x{\sec }^{4}xdx$ equal to ?

$\int \frac{\sqrt{x}}{\sqrt{\left(a^3-x^3\right)}}dx$ is equal to

What is $\int e^x\left(\sqrt{x}+\frac{1}{2\sqrt{x}}\right)dx$ is equal to ?

$\int \frac{{\left(x^4-x\right)}^{{\displaystyle \frac{1}{4}}}}{x^5}dx$ is equal to

If $\int \frac{dx}{(x-2{)}^2\left(x-3\right)}=\frac{C_1}{x-2}+C_2\log \left(\frac{x-2}{x-3}\right)+C_3$, where $C_3$ is constant of integration, then

What is $\int \frac{e^x\left(1+x\right)}{{\cos }^2\left(x{e}^x\right)}dx$ equal to ?

If $xf\left(x\right)=3f^2\left(x\right)+2,$ then $\int \frac{2x^2-12xf\left(x\right)+f\left(x\right)}{\left(6f\left(x\right)-x\right){\left(x^2-f\left(x\right)\right)}^2}dx$ equals

$4\int \frac{\sqrt{a^6+x^8}}{x}dx$ is equal to

If $\int x\frac{\ln \left(x+\sqrt{1+x^2}\right)}{\sqrt{1+x^2}}dx$  $=a\sqrt{1+x^2}\ln \left(x+\sqrt{1+x^2}\right)+bx+c,$ then