What is the integral of ln(5^x) ?

The integral of ln(5^x) is 1/2 ln(5)x^2+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of ln(5^x)

\int ln (5^{x}) d x

\frac{1}{2} ln (5) x^{2} + C

Solution steps

\int ln (5^{x}) d x

Apply log rule

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

assuming

x \geq 0

= \int ln (5) x d x

Take the constant out:

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

= ln (5) \cdot \int x d x

Apply the Power Rule

= ln (5) \frac{x ^{2}}{2}

Simplify

ln (5) \frac{x ^{2}}{2} : \frac{1}{2} ln (5) x^{2}

= \frac{1}{2} ln (5) x^{2}

Add a constant to the solution

= \frac{1}{2} ln (5) x^{2} + C