What is the integral of x^5ln(x^3+1) ?

The integral of x^5ln(x^3+1) is 1/12 (2x^6ln(x^3+1)-x^6+2x^3-2ln(x^3+1)+3)+C

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integral of x^5ln(x^3+1)

\int x^{5} ln (x^{3} + 1) d x

\frac{1}{12} (2 x^{6} ln (x^{3} + 1) - x^{6} + 2 x^{3} - 2 ln (x^{3} + 1) + 3) + C

Solution steps

\int x^{5} ln (x^{3} + 1) d x

Apply u-substitution

: \int \frac{( u - 1 ) ln ( u )}{3} d u

= \int \frac{( u - 1 ) ln ( u )}{3} d u

Take the constant out:

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

= \frac{1}{3} \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}{3} (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}{3} (ln (u) (\frac{u ^{2}}{2} - u) - (\frac{u ^{2}}{4} - u))

Substitute back

u = x^{3} + 1

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

Simplify

\frac{1}{3} (ln (x^{3} + 1) (\frac{( x ^{3} + 1 ) ^{2}}{2} - (x^{3} + 1)) - (\frac{( x ^{3} + 1 ) ^{2}}{4} - (x^{3} + 1))) : \frac{1}{12} (2 x^{6} ln (x^{3} + 1) - x^{6} + 2 x^{3} - 2 ln (x^{3} + 1) + 3)

= \frac{1}{12} (2 x^{6} ln (x^{3} + 1) - x^{6} + 2 x^{3} - 2 ln (x^{3} + 1) + 3)

Add a constant to the solution

= \frac{1}{12} (2 x^{6} ln (x^{3} + 1) - x^{6} + 2 x^{3} - 2 ln (x^{3} + 1) + 3) + C

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What is the integral of x^5ln(x^3+1) ?
The integral of x^5ln(x^3+1) is 1/12 (2x^6ln(x^3+1)-x^6+2x^3-2ln(x^3+1)+3)+C