Aus Online Mathematik Brückenkurs 2

If we multiply the factors in the integrand together and use the power laws,

\displaystyle \begin{align} \int e^{2x}\bigl(e^x+1\bigr)\,dx &= \int\bigl(e^{2x}e^{x} + e^{2x}\bigr)\,dx\\[5pt] &= \int\bigl(e^{2x+x} + e^{2x}\bigr)\,dx\\[5pt] &= \int{\bigl(e^{3x} + e^{2x}\bigr)}\,dx\,, \end{align}

we obtain a standard integral with two terms of the type \displaystyle e^{ax}, where \displaystyle a is a constant. The indefinite integral is therefore

\displaystyle \int \bigl(e^{3x}+e^{2x}\bigr)\,dx = \frac{e^{3x}}{3} + \frac{e^{2x}}{2} + C\,,

where \displaystyle C is an arbitrary constant.