Evaluate the following integrals :
(i) `int x^(15)dx` (ii) `int x^(-3//2)dx` (iii) `int(3x^(-7)+x^(-1))dx`
(iv) `int(sqrt(x)+(1)/sqrt(x))^(2)dx` (v) `int(x+(1)/(x))dx` (vi) `int((a)/(x^(2))+(b)/(x))dx` (a and b are constant)

Answer

(i) `(x^(16))/(16) + c`
(ii) `-2x^(-1//2) +c`
(iii) `-(x^(-6))/(2)+ log_(e)x+c`
(iv) `(x^(2))/(2) + 2x + log_ex+c`
(v) `(x^(2))/(2) +log_ex+c`
(vi) `-(a)/(x) + blog_ex+c `

Explanation

(i) `int x^(15)dx = (x^(15+1))/(15+1)+C= (x^(16))/(16)+C`
(ii) `int x^(-3//2) dx = (x^((3)/(2)+1))/(- (3)/(2) + 1) +C= - 2x^(- (1)/(2)) + C`
(iii) `int (3x^(-7) + x^(-1)) dx = (3x^(-6))/(-6) + log_ex+C`
`" "= (-x^(-6))/(2) + log_ex+ C`
(iv) `int (sqrtx+ (1)/(sqrtx))^(2) dx`
`= int (x+ (1)/(x) + 2)dx = (x^(2))/(2) + log_ex+ 2x+ C`
(v) ` int (x+ (1)/(x))dx =(x^(2))/(2) + log_ex + C`
(vi) `int ((a)/(x^(2))+ (b)/(x)) dx = int (ax^(-2) + bx^(-1)) dx `
`= (ax^(-2+1))/(-2+1)+ blog_ex+ C= -(a)/(x) + blog_ex+ C`