Question

How do you integrate `int sqrtxlnx` by integration by parts method?

Answer 1

`intsqrt(x)ln(x)dx=2/3x^(3/2)(ln(x)-2/3)+C`

Explanation:

Using integration by parts:

Let `u = ln(x) => du = 1/xdx`
and `dv = x^(1/2)dx => v = 2/3x^(3/2)`

Using the integration by parts formula `intudv = uv-intvdu`

`intsqrt(x)ln(x)dx = intln(x)x^(1/2)dx`

`=2/3x^(3/2)ln(x)-int2/3x^(3/2)*1/xdx`

`=2/3x^(3/2)ln(x)-2/3intx^(1/2)dx`

`=2/3x^(3/2)ln(x)-2/3(2/3x^(3/2))+C`

`=2/3x^(3/2)(ln(x)-2/3)+C`