What is the integral of tln(t+5) ?

The integral of tln(t+5) is ln(t+5)(1/2 (t+5)^2-5(t+5))-1/4 (t^2-10t-75)+C

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integral of tln(t+5)

\int t ln (t + 5) d t

ln (t + 5) (\frac{1}{2} (t + 5)^{2} - 5 (t + 5)) - \frac{1}{4} (t^{2} - 10 t - 75) + C

Solution steps

\int t ln (t + 5) d t

Apply u-substitution

= \int (u - 5) ln (u) d u

Apply Integration By Parts

= ln (u) (\frac{u ^{2}}{2} - 5 u) - \int \frac{1}{2} (u - 10) d u

\int \frac{1}{2} (u - 10) d u = \frac{1}{2} (\frac{u ^{2}}{2} - 10 u)

= ln (u) (\frac{u ^{2}}{2} - 5 u) - \frac{1}{2} (\frac{u ^{2}}{2} - 10 u)

Substitute back

u = t + 5

= ln (t + 5) (\frac{( t + 5 ) ^{2}}{2} - 5 (t + 5)) - \frac{1}{2} (\frac{( t + 5 ) ^{2}}{2} - 10 (t + 5))

Simplify

ln (t + 5) (\frac{( t + 5 ) ^{2}}{2} - 5 (t + 5)) - \frac{1}{2} (\frac{( t + 5 ) ^{2}}{2} - 10 (t + 5)) : ln (t + 5) (\frac{1}{2} (t + 5)^{2} - 5 (t + 5)) - \frac{1}{4} (t^{2} - 10 t - 75)

= ln (t + 5) (\frac{1}{2} (t + 5)^{2} - 5 (t + 5)) - \frac{1}{4} (t^{2} - 10 t - 75)

Add a constant to the solution

= ln (t + 5) (\frac{1}{2} (t + 5)^{2} - 5 (t + 5)) - \frac{1}{4} (t^{2} - 10 t - 75) + C

What is the integral of tln(t+5) ?
The integral of tln(t+5) is ln(t+5)(1/2 (t+5)^2-5(t+5))-1/4 (t^2-10t-75)+C