anti-derivative

[scheely]

Hi - was looking for help in regards determining anti-derivatives.
I am asked to find a derivative of f(x) = x^2ln(2x), finding f'(x); and then hence the anti derivative integer xln(2x)dx. (where ln is log e).

[fun_jirachi]

Welcome to the forums!

Now, the derivative of \(x^2 \ln (2x) = x^2 \times \frac{2}{2x} + 2x\ln (2x) = x + 2x\ln (2x)\) by the product rule.

Hence, the antiderivative of \(x\ln (2x)\) will be equal to \(\frac{1}{2}\int x + 2x\ln (2x) - x \ dx\).
Evaluating this integral now:


Note how we use the fundamental theorems of calculus and our previous work + how we manipulate the expression provided in the question into a more familiar form that allows us to employ said method. Try a few more of these and it should bed in pretty handily.

Hope this helps