What is the integral of x^(7/2) * ln(x) dx? Thanks
To find the integral of x^(7/2) * ln(x) dx, we can use integration by parts. The general formula for integration by parts is:
∫ u dv = u * v - ∫ v du
In this case, let's assign u = ln(x) and dv = x^(7/2) dx. Taking the derivatives and antiderivatives, we have:
du = (1/x) dx
v = (2/9) x^(9/2)
Now, applying the formula:
∫ x^(7/2) * ln(x) dx = u * v - ∫ v du
Substituting the values, we get:
∫ x^(7/2) * ln(x) dx = ln(x) * (2/9) x^(9/2) - ∫ (2/9) x^(9/2) (1/x) dx
Simplifying further, we have:
∫ x^(7/2) * ln(x) dx = (2/9) x^(9/2) ln(x) - (2/9) ∫ x^(7/2) dx
Integrating the remaining term:
∫ x^(7/2) dx = (2/9) * (2/9+2) x^(9/2 + 1) + C
Combining these results, we get the final answer:
∫ x^(7/2) * ln(x) dx = (2/9) x^(9/2) ln(x) - (2/9) * (2/11) x^(11/2) + C
where C represents the constant of integration.