# How do you find the integral #ln(x) x^(3/2) dx#?

To carry out component integration:

That's not too bad, nice.

By signing up, you agree to our Terms of Service and Privacy Policy

To find the integral (\int \ln(x) x^{3/2} , dx), we can use integration by parts. Let (u = \ln(x)) and (dv = x^{3/2} , dx). Then, we have (du = \frac{1}{x} , dx) and (v = \frac{2}{5}x^{5/2}).

Applying the integration by parts formula:

[ \int u , dv = uv - \int v , du ]

we get:

[ \int \ln(x) x^{3/2} , dx = \frac{2}{5}x^{5/2} \ln(x) - \int \frac{2}{5}x^{5/2} \frac{1}{x} , dx ]

Simplify the integral:

[ = \frac{2}{5}x^{5/2} \ln(x) - \frac{2}{5} \int x^{3/2} , dx ]

Now, integrate (\int x^{3/2} , dx) to get:

[ = \frac{2}{5}x^{5/2} \ln(x) - \frac{2}{5} \left(\frac{2}{5}x^{5/2}\right) + C ]

So, the integral of (\ln(x) x^{3/2} , dx) is:

[ \frac{2}{5}x^{5/2} \ln(x) - \frac{4}{25}x^{5/2} + C ]

By signing up, you agree to our Terms of Service and Privacy Policy

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7