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

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

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integral of xln(x^2+3)

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

\int 1 d u = u

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

Substitute back

u = x^{2} + 3

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

Expand

(x^{2} + 3) ln (x^{2} + 3) - (x^{2} + 3) : x^{2} ln (x^{2} + 3) + 3 ln (x^{2} + 3) - x^{2} - 3

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

Add a constant to the solution

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

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