# How do you integrate #int tln(t+1)# by parts?

integration by parts formula

for

divide the improper fraction out

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To integrate ( \int t \ln(t+1) ) by parts, you can use the formula:

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

Let ( u = \ln(t+1) ) and ( dv = t , dt ). Then, ( du = \frac{1}{t+1} , dt ) and ( v = \frac{1}{2} t^2 ).

Now apply the formula:

[ \int t \ln(t+1) , dt = \frac{1}{2} t^2 \ln(t+1) - \int \frac{1}{2} t^2 \cdot \frac{1}{t+1} , dt ]

Simplify the integral:

[ \int \frac{1}{2} t^2 \cdot \frac{1}{t+1} , dt = \frac{1}{2} \int \frac{t^2}{t+1} , dt ]

This integral can be solved by substitution or partial fractions, depending on the preferred method. Once you find the antiderivative of ( \frac{t^2}{t+1} ), you can combine it with the previous result to obtain the final answer.

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