What is the integral of xln(sqrt(x+1)) ?

The integral of xln(sqrt(x+1)) is 1/8 (2x^2ln(x+1)-x^2+2x-2ln(x+1)+3)+C

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integral of xln(sqrt(x+1))

\int x ln (x + 1) d x

\frac{1}{8} (2 x^{2} ln (x + 1) - x^{2} + 2 x - 2 ln (x + 1) + 3) + C

Solution steps

\int x ln (x + 1) d x

Simplify

ln (x + 1) : \frac{1}{2} ln (x + 1)

= \int \frac{1}{2} x ln (x + 1) d x

Take the constant out:

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

= \frac{1}{2} \cdot \int x ln (x + 1) d x

Apply u-substitution

: \int (u - 1) ln (u) d u

= \frac{1}{2} \cdot \int (u - 1) ln (u) d u

Apply Integration By Parts

: ln (u) (\frac{u ^{2}}{2} - u) - \int \frac{1}{2} (u - 2) d u

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

\int \frac{1}{2} (u - 2) d u = \frac{u ^{2}}{4} - u

= \frac{1}{2} (ln (u) (\frac{u ^{2}}{2} - u) - (\frac{u ^{2}}{4} - u))

Substitute back

u = x + 1

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

Simplify

\frac{1}{2} (ln (x + 1) (\frac{( x + 1 ) ^{2}}{2} - (x + 1)) - (\frac{( x + 1 ) ^{2}}{4} - (x + 1))) : \frac{1}{8} (2 x^{2} ln (x + 1) - x^{2} + 2 x - 2 ln (x + 1) + 3)

= \frac{1}{8} (2 x^{2} ln (x + 1) - x^{2} + 2 x - 2 ln (x + 1) + 3)

Add a constant to the solution

= \frac{1}{8} (2 x^{2} ln (x + 1) - x^{2} + 2 x - 2 ln (x + 1) + 3) + C

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What is the integral of xln(sqrt(x+1)) ?
The integral of xln(sqrt(x+1)) is 1/8 (2x^2ln(x+1)-x^2+2x-2ln(x+1)+3)+C