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

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

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integral of ln(5^u)

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

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

Solution steps

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

Apply log rule

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

assuming

x \geq 0

= \int ln (5) u d u

Take the constant out:

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

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

Apply the Power Rule

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

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

Simplify

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

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

Add a constant to the solution

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

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