What is the integral of ln(x^n) ?

The integral of ln(x^n) is nxln(x)-nx+C

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integral of ln(x^n)

\int ln (x^{n}) d x

n x ln (x) - n x + C

Solution steps

\int ln (x^{n}) d x

Apply Integration By Parts

: x ln (x^{n}) - \int n d x

= x ln (x^{n}) - \int n d x

\int n d x = n x

= x ln (x^{n}) - n x

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

assuming

x \geq 0

= n x ln (x) - n x

Add a constant to the solution

= n x ln (x) - n x + C

\int \frac{sec ^{4} ( x )}{tan ^{5} ( x )} d x

(\frac{3}{10}, - \frac{1}{2}), (\frac{1}{5}, \frac{1}{5})

\frac{\partial}{\partial x} (y^{3} sin (5 x))

\frac{z}{z + \frac{d}{z}}