Question

What is the integral of x*sin(sqrt.(x^2 + 4))/sqrt.(x^2 + 4) ?

Answer 1

`I=-cos(sqrt(x^2+4))+C`

Explanation:

We want to integrate

`I=intx*sin(sqrt(x^2+4))/sqrt(x^2+4)dx`

Make a substitution `color(blue)(u=x^2+4=>du=2xdx`

`I=intx*sin(sqrt(u))/sqrt(u)*1/(2x)du`

`color(white)(I)=1/2intsin(sqrt(u))/sqrt(u)du`

Make a substitution `color(blue)(s=sqrt(u)=>ds=1/(2s)du`

`I=1/2intsin(s)/s*2sds`

`color(white)(I)=intsin(s)ds`

`color(white)(I)=-cos(s)+C`

Substitute back `color(blue)(s=sqrt(u)` and `color(blue)(u=x^2+4`

`I=-cos(sqrt(x^2+4))+C`