Integration problems?

Question

1) ∫ (x^2)(ln x) dx

2) ∫ 1/( (sqrt x)(1+x) ) dx

Answer 1

1) ∫ (x^2)(ln x) dx

You have to solve this using integration by parts.

Let u = ln(x). dv = x^2 dx
du = (1/x) dx. v = (1/3)x^3

Therefore, our integral becomes

(1/3)x^3 ln(x) - ∫ [(1/x)(1/3)x^3] dx

Merge everything together in the integral,

(1/3)x^3 ln(x) - ∫ [(x^3)/(3x)] dx

Now, note that the x in the denominator cancels out, while reducing the power of the integral by 1.

(1/3)x^3 ln(x) - ∫ [(x^2)/3] dx

Now, we can pull out the (1/3) as a constant outside of the integral.

(1/3)x^3 ln(x) - (1/3) ∫ x^2 dx

We can now solve this normally, using the reverse power rule.

(1/3)x^3 ln(x) - (1/3) [(1/3)x^3] + C

Reducing this slightly,

(1/3)x^3 ln(x) - (1/9)x^3 + C