Question

Integral of 1/(route 2x+5)+(route 2x-3)?

• 1/`sqrt(2x+5)`+`sqrt(2x-3`

Answer 1

`I=1/24((2x+5)^(3/2)-(2x-3)^(3/2))+C`

Explanation:

We want to solve

`I=int1/(sqrt(2x+5)+sqrt(2x-3))dx`

Rationalize the denominator of the integrand

`I=int1/(sqrt(2x+5)+sqrt(2x-3))*(sqrt(2x+5)-sqrt(2x-3))/(sqrt(2x+5)-sqrt(2x-3))dx`

`color(white)(I)=int(sqrt(2x+5)-sqrt(2x-3))/(2x+5-(2x-3))dx`

`color(white)(I)=1/8intsqrt(2x+5)-sqrt(2x-3)dx`

Make a substitution `color(brown)(u=2x=>du=2dx`

`I=1/16intsqrt(u+5)-sqrt(u-3)dx`

`color(white)(I)=1/16(2/3(u+5)^(3/2)-2/3(u-3)^(3/2))+C`

`color(white)(I)=1/24((u+5)^(3/2)-(u-3)^(3/2))+C`

Substitute back `u=2x`

`I=1/24((2x+5)^(3/2)-(2x-3)^(3/2))+C`