What is the integral of x*ln(1+x) ?

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

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integral of x*ln(1+x)

\int x \cdot ln (1 + x) d x

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

Solution steps

\int x ln (1 + x) d x

Apply u-substitution

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

= \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

= 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

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

Substitute back

u = 1 + x

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

Simplify

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

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

Add a constant to the solution

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

x^{3} - 5 x^{2} - x + 5, 1, 5

\int e^{- 2 x} sin (\frac{x}{2}) d x

f (x) = arctan (\frac{2}{x})

\int x \cdot cos (k x) d x

- 10 x^{4} + x^{3}

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