What is the integral of x^2ln(x^3+1) ?

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

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integral of x^2ln(x^3+1)

\int x^{2} ln (x^{3} + 1) d x

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

Solution steps

\int x^{2} ln (x^{3} + 1) d x

Apply u-substitution

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

= \int \frac{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 ln (u) d u

Apply Integration By Parts

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

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

\int 1 d u = u

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

Substitute back

u = x^{3} + 1

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

Expand

(x^{3} + 1) ln (x^{3} + 1) - (x^{3} + 1) : x^{3} ln (x^{3} + 1) + ln (x^{3} + 1) - x^{3} - 1

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

Add a constant to the solution

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

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