What is the integral of x*ln(5+x) ?

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

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integral of x*ln(5+x)

\int x \cdot ln (5 + x) d x

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

Solution steps

\int x ln (5 + x) 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 = 5 + x

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

Simplify

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

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

Add a constant to the solution

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

\int \frac{π}{2} - 5 x^{- 0.5} d x

\frac{d ^{3}}{d x ^{3}} (x^{4} - 8 x^{3})

\frac{d}{d x} (e^{c o s (x + 3)})

- 8 t^{2} y^{^{''}} - 4 t (t - 4) y^{^{'}} + 4 (t - 4) y = 0

y^{2} = x, x - 2 y = 3

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