Find the following indefinite integral using the method of integration by parts.

Question

Find the following indefinite integral using the method of integration by parts. xlog(6x)dx Recall that integration by parts relies on the "reverse" of the Product rule of derivatives. This can be written, f(x)g'(x)dx = f(x)g(x)- f'(x)g{x)dx What do you choose for f(x)? f(x) = Enter just an expression in x. What do you choose for g'(x)? g'(x) = Enter just an expression in x. What then is f'(x) the derivative of f(x)? f(x) = d / dxf(x) = Enter just an expression in x. What then is g(x) an antiderivative of g'(x)? g(x) = g'(x) dx = Enter just an expression in x. Note here that you do not need a constant of integration. What is the indefinite integral? f(x) g'(x) dx = Enter just an expression, and be careful of the signs.

Explanation / Answer

f(x) = log(6x)

g'(x) = x

f'(x) = 1/x

g(x) = x2/2

f(x).g(x) = x2/2 *log(6x) - integral(1/x)*(x2/2)dx

= x2/2 *log(6x) - (x2/4)

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