Skip to Main Content
It looks like you're using Internet Explorer 11 or older. This website works best with modern browsers such as the latest versions of Chrome, Firefox, Safari, and Edge. If you continue with this browser, you may see unexpected results.


Integrals of Exponential Form

We know that the derivative of the exponential function \(de^u/dx=e^u\, du/dx\). Reversing the differential, the integral of an exponential function is

\[\int e^udu=e^u+C\]

Example 1 Integrate \[\int 6x^2e^{x^3}dx\]

Solution: let \(u=x^3\),

\begin{align} du&= 3x^2dx \\ \frac{du}{3} &=x^2dx \end{align}

We can substitute into our original integral and get,

\begin{align} 2\int e^udu &=2e^u +C \\ &=2e^{x^3}+C \end{align}

Example 2: Integrate \[\int \left(e^x-e^{-x}\right)^2dx\]

Solution: We can simplify the expression in the integral to help us integrate.

\begin{align} \left(e^x-e^{-x}\right)^2 &= e^{2x}-2e^xe^{-x}+e^{-2x}\\ &=e^{2x}-2+e^{-2x} \end{align}


\[ \int \left(e^{2x}-2+e^{-2x}\right) dx = \frac{e^{2x}}{2} -2x - \frac{e^{-2x}}{2} +C \]

Example 3: Find the equation of the curve for which \(dy/dx=\sqrt{e^{x+3}}\) if the curve passes through \((1,0)\).

Solution: First, we integrate to find the general equation form

\begin{align} y&=\int e^{\frac{x+3}{2}}dx \\ &= 2e^{\frac{x+3}{2}}+C\end{align}

Now we solve for C using the point \((1,0)\)

\begin{align} 0&=2e^{\frac{1+3}{2}}+C \\ -2e^2&=C \end{align}

Thus, the equation of the curve is


Creative Commons License
Designed by Matthew Cheung. This work is licensed under a Creative Commons Attribution 4.0 International License.

chat loading...