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Wize AP Calculus (AB) Textbook > Integration & Accumulation of Change

Antiderivatives of Exponential Functions

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Antiderivatives (Indefinite Integral) of Exponential Functions

Let kk be a nonzero constant. Then we have the following indefinite integral rules:
∫ekxdx=ekxk+C\displaystyle\boxed{\int e^{kx}dx=\dfrac{e^{kx}}{k}+C}

∫akxdx=akxkln⁡a+C, (for a>0,a≠1)\displaystyle\boxed{\displaystyle \int a^{kx}dx=\dfrac{a^{kx}}{k\ln a}+C},\ \text{(for}\ a>0,a\neq1)

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Example: Exponential Indefinite Integrals

Evaluate the following Indefinite Integral

∫(ex)9dx\displaystyle \int (e^x)^9dx

∫(ex)9dx\displaystyle \int (e^x)^9dx

=∫e9xdx=\displaystyle \int e^{9x}dx

=e9x9+C\displaystyle =\frac{e^{9x}}{9}+C
Evaluate the following indefinite integral

∫(e3−e3x+3x)dx\displaystyle \int( e^3-e^{3x}+3^x )dx

Evaluate the following indefinite integral

∫7(x2+1)dx\displaystyle \int 7^{\left(\displaystyle \frac{x}{2}+1\right)}dx