Question

What will be the Evaluate the following integrals ? (1) ∫ (a/x^2 + b/x) dx (2) ∫ (3x^-7+x^-1) (3) ∫ (√x+1/√x)^2 dx

Answer 1

(1) `int \ a/x^2+b/x \ dx = -a/x + bln|x| + c`

(2) `int \ 3x^(-7)+x^(-1) \ dx = -1/(2x^6) + ln|x| + c`

(3) `int \ (sqrt(x) +1/sqrt(x))^2 \ dx =x^2/2+2x + ln|x| + c`

Explanation:

We use the power rule:

`int \ x^n \ dx = x^(n+1)/(n+1) \ \ \ (+c) \ \ \ AA x in RR, x!= -1`

And the standard result:

`int \ 1/x \ dx = lnx \ \ \ \ (+c)`

So that:

Part (1):

`I_1 = int \ a/x^2+b/x \ dx`

`\ \ \ = int \ ax^(-2)+b/x \ dx`

`\ \ \ = ax^(-1)/(-1) + bln|x| + c`

`\ \ \ = -a/x + bln|x| + c`

Part (2):

`I_2 = int \ 3x^(-7)+x^(-1) \ dx`

`\ \ \ = 3x^(-6)/(-6) + ln|x| + c`

`\ \ \ = -1/(2x^6) + ln|x| + c`

Part (3):

`I_3 = int \ (sqrt(x) +1/sqrt(x))^2 \ dx`

`\ \ \ = int \ (sqrt(x))^2 + 2 sqrt(x)1/sqrt(x) + (1/sqrt(x))^2 \ dx`

`\ \ \ = int \ x + 2 + 1/x \ dx`

`\ \ \ = x^2/2+2x + ln|x| + c`