Integral of e to the Power of a Function

The integration of $e$ to the power $x$ of a function is a general formula of exponential functions and this formula needs a derivative of the given function. This formula is important in integral calculus.

The integration of e to the power x of a function is of the form

Now consider

Using the derivative formula $\frac{d}{{dx}}{e^{f\left( x \right)}} = {e^{f\left( x \right)}}f'\left( x \right)$, we have

Integrating both sides of equation (i) with respect to $x$, we have

Since integration and differentiation are reverse processes to each other, the integral sign $\int {}$ and $\frac{d}{{dx}}$ on the right side will cancel each other out, i.e.

Example: Evaluate the integral $\int {\frac{{{e^{{{\sin }^{ - 1}}x}}}}{{\sqrt {1 - {x^2}} }}dx}$ with respect to $x$

We have integral

Here $f\left( x \right) = {\sin ^{ - 1}}x$, and $d\left( x \right) = \frac{1}{{\sqrt {1 - {x^2}} }}$, so we can write it as

Using the integration formula $\int {{e^{f\left( x \right)}}f'\left( x \right)dx = } {e^{f\left( x \right)}} + c$, we have