What is the integral of (ln(x^{25}))/x ?

The integral of (ln(x^{25}))/x is 25/2 ln^2(x)+C

integral of (ln(x^{25}))/x

\int \frac{ln ( x ^{25} )}{x} d x

\frac{25}{2} ln^{2} (x) + C

Solution steps

\int \frac{ln ( x ^{25} )}{x} d x

Simplify

ln (x^{25}) : 25 ln (x)

= \int \frac{25 ln ( x )}{x} d x

Take the constant out:

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

= 25 \cdot \int \frac{ln ( x )}{x} d x

Apply u-substitution

= 25 \cdot \int u d u

Apply the Power Rule

= 25 \cdot \frac{u ^{2}}{2}

Substitute back

u = ln (x)

= 25 \cdot \frac{ln ^{2} ( x )}{2}

Simplify

25 \cdot \frac{ln ^{2} ( x )}{2} : \frac{25}{2} ln^{2} (x)

= \frac{25}{2} ln^{2} (x)

Add a constant to the solution

= \frac{25}{2} ln^{2} (x) + C

\frac{\partial}{\partial y} (x^{5} + 2^{y} + x^{y})

x \to 2 - lim (x + 2)

y^{^{''''}} + 20 y^{^{''}} + 100 y = 0

y^{^{'}} = e^{5 x + 3 y}

f (t) = cos^{3} (t)

What is the integral of (ln(x^{25}))/x ?
The integral of (ln(x^{25}))/x is 25/2 ln^2(x)+C