What is the integral of atln(t+b) ?

The integral of atln(t+b) is a(1/2 t^2ln(t+b)-1/4 (t^2-2bt+2b^2ln| t/b+1|-3b^2))+C

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integral of atln(t+b)

\int a t ln (t + b) d t

a (\frac{1}{2} t^{2} ln (t + b) - \frac{1}{4} (t^{2} - 2 b t + 2 b^{2} ln \frac{t}{b} + 1 - 3 b^{2})) + C

Solution steps

\int a t ln (t + b) d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= a \cdot \int t ln (t + b) d t

Apply Integration By Parts

: \frac{1}{2} t^{2} ln (t + b) - \int \frac{t ^{2}}{2 ( t + b )} d t

= a (\frac{1}{2} t^{2} ln (t + b) - \int \frac{t ^{2}}{2 ( t + b )} d t)

\int \frac{t ^{2}}{2 ( t + b )} d t = \frac{1}{4} (t^{2} - 2 b t + 2 b^{2} ln \frac{t}{b} + 1 - 3 b^{2})

= a (\frac{1}{2} t^{2} ln (t + b) - \frac{1}{4} (t^{2} - 2 b t + 2 b^{2} ln \frac{t}{b} + 1 - 3 b^{2}))

Add a constant to the solution

= a (\frac{1}{2} t^{2} ln (t + b) - \frac{1}{4} (t^{2} - 2 b t + 2 b^{2} ln \frac{t}{b} + 1 - 3 b^{2})) + C

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What is the integral of atln(t+b) ?
The integral of atln(t+b) is a(1/2 t^2ln(t+b)-1/4 (t^2-2bt+2b^2ln| t/b+1|-3b^2))+C