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# Evaluation of Simple Integrals of the Following Types and Problems

#### notes

Integral of the type:
int e^x [ f(x) + f'(x)] dx
we have I = int e^x[f(x) + f'(x)] dx
= int e^x f(x) dx + int e^x f'(x) dx
= I_1 + int e^x f'(x) dx , "where"  I_1 = int e^x f(x) dx         ...(1)
Taking f(x) and e^x as the first function and second function, respectively, in I_1 and integrating it by parts, we have I_1
= f(x) e^x - int f'(x) e^x dx +C
Substituting I_1 in (1), we get
I = e^x f(x) - int f'(x) e^x dx + int e^x f'(x) dx +C = e^x f(x) + C
Thus , int e^x [f(x) + f'(x)] dx = e^x f(x) + C

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

ii) int sqrt (x^2 + a^2) dx
= 1/2 x sqrt (x^2 + a^2) + a^2/2 log |x + sqrt (x^2 +a^2)| +C

iii) int sqrt (a^2 - x^2)dx = 1/2 x sqrt (a^2-x^2) + a^2/2  sin^(-1)  x/a+C