Evaluation of Simple Integrals of the Following Types and Problems

Integral of the type:
${\displaystyle \int e^x[f\left(x\right)+f\prime \left(x\right)]dx}$
we have I = ${\displaystyle \int e^x[f\left(x\right)+f\prime \left(x\right)]dx}$ 
= ${\displaystyle \int e^{x}f\left(x\right)dx+\int e^{x}f\prime \left(x\right)dx}$
= ${\displaystyle I_1+\int e^{x}f\prime \left(x\right)dx,\text{where} I_1=\int e^{x}f\left(x\right)dx}$         ...(1)
Taking f(x) and ${\displaystyle e^x}$ as the first function and second function, respectively, in ${\displaystyle I_1}$ and integrating it by parts, we have ${\displaystyle I_1}$
= ${\displaystyle f\left(x\right)e^x-\int f\prime \left(x\right)e^{x}dx+C}$
Substituting ${\displaystyle I_1}$ in (1), we get 
I = ${\displaystyle e^{x}f\left(x\right)-\int f\prime \left(x\right)e^{x}dx+\int e^{x}f\prime \left(x\right)dx+C=e^{x}f\left(x\right)+C}$
Thus , ${\displaystyle \int e^x[f\left(x\right)+f\prime \left(x\right)]dx=e^{x}f\left(x\right)+C}$

Integrals of some more types :
Some special types of standard integrals based on the technique of integration  by parts :
i) ${\displaystyle \int \sqrt{x^2-a^2}dx}$ 
${\displaystyle I=\int \sqrt{x^2-a^2}dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\log |x+\sqrt{x^2-a^2}|+C}$

ii) ${\displaystyle \int \sqrt{x^2+a^2}dx}$
= ${\displaystyle \frac{1}{2}x\sqrt{x^2+a^2}+\frac{a^2}{2}\log |x+\sqrt{x^2+a^2}|+C}$

iii) ${\displaystyle \int \sqrt{a^2-x^2}dx=\frac{1}{2}x\sqrt{a^2-x^2}+\frac{a^2}{2} {\sin }^{-1} \frac{x}{a}+C}$
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