What is the integral of xln(x+5) ?

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

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integral of xln(x+5)

\int x ln (x + 5) d x

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

Solution steps

\int x ln (x + 5) d x

Apply u-substitution

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

= \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

= 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 = x + 5

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

Simplify

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

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (ln (x^{2} + y^{5}))

\int ln (x + 2) d x

\int 7 x^{3} e^{x} d x

f (x) = x^{2} - 3 x + 4, at x = 1

\frac{d y}{d t} + 2 t y = 2 t e^{- t^{2}}

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