What is the integral of (xln^5(x^2+3))/(x^2+3) ?

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

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integral of (xln^5(x^2+3))/(x^2+3)

\int \frac{x ln ^{5} ( x ^{2} + 3 )}{x ^{2} + 3} d x

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

Solution steps

\int \frac{x ln ^{5} ( x ^{2} + 3 )}{x ^{2} + 3} d x

Apply u-substitution

: \int \frac{ln ^{5} ( u )}{2 u} d u

= \int \frac{ln ^{5} ( u )}{2 u} 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 \frac{ln ^{5} ( u )}{u} d u

Apply u-substitution

: \int v^{5} d v

= \frac{1}{2} \cdot \int v^{5} d v

Apply the Power Rule

: \frac{v ^{6}}{6}

= \frac{1}{2} \cdot \frac{v ^{6}}{6}

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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