What is the integral of ((ln(x^{15})))/x ?

The integral of ((ln(x^{15})))/x is 15/2 ln^2(x)+C

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integral of ((ln(x^{15})))/x

\int \frac{( ln ( x ^{15} ) )}{x} d x

\frac{15}{2} ln^{2} (x) + C

Solution steps

\int \frac{ln ( x ^{15} )}{x} d x

Simplify

ln (x^{15}) : 15 ln (x)

= \int \frac{15 ln ( x )}{x} d x

Take the constant out:

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

= 15 \cdot \int \frac{ln ( x )}{x} d x

Apply u-substitution

: \int u d u

= 15 \cdot \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= 15 \cdot \frac{u ^{2}}{2}

Substitute back

u = ln (x)

= 15 \cdot \frac{ln ^{2} ( x )}{2}

Simplify

15 \cdot \frac{ln ^{2} ( x )}{2} : \frac{15}{2} ln^{2} (x)

= \frac{15}{2} ln^{2} (x)

Add a constant to the solution

= \frac{15}{2} ln^{2} (x) + C

5 x e^{x}

\frac{2 x ^{3} - 4 x ^{2} + 3}{x}

\int_{1}^{7} \frac{5}{x} d x

x = y^{2} + 6 y, y = x

n = 0 \sum \infty x^{n + 5}

What is the integral of ((ln(x^{15})))/x ?
The integral of ((ln(x^{15})))/x is 15/2 ln^2(x)+C