What is the integral of ln(2^x) ?

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

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integral of ln(2^x)

\int ln (2^{x}) d x

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

Solution steps

\int ln (2^{x}) d x

Apply Integration By Parts

= ln (2) x^{2} - \int ln (2) x d x

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

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

Simplify

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

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

Add a constant to the solution

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