# Integral of Constant to the Power of a Function

The integration of any constant power of a function is a general formula of exponential functions, and this formula needs the derivative of the given function. This formula is important in integral calculus.

The integration of any constant power of a function is of the form
$\int {{a^{f\left( x \right)}}f’\left( x \right)dx = } \frac{{{a^{f\left( x \right)}}}}{{\ln a}} + c$

Now consider
$\frac{d}{{dx}}\left[ {\frac{{{a^{f\left( x \right)}}}}{{\ln a}} + c} \right] = \frac{1}{{\ln a}}\frac{d}{{dx}}{a^{f\left( x \right)}} + \frac{d}{{dx}}c$

Using the derivative formula $\frac{d}{{dx}}{a^{f\left( x \right)}} = {a^{f\left( x \right)}}\ln af’\left( x \right)$, we have
$\begin{gathered} \frac{d}{{dx}}\left[ {\frac{{{a^{f\left( x \right)}}}}{{\ln a}} + c} \right] = \frac{1}{{\ln a}}{a^{f\left( x \right)}}\ln af’\left( x \right) + 0 \\ \Rightarrow {a^{f\left( x \right)}}f’\left( x \right) = \frac{d}{{dx}}\left[ {\frac{{{a^{f\left( x \right)}}}}{{\ln a}} + c} \right] \\ \Rightarrow {a^{f\left( x \right)}}f’\left( x \right)dx = d\left[ {\frac{{{a^{f\left( x \right)}}}}{{\ln a}} + c} \right]\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ \end{gathered}$

Integrating both sides of equation (i) with respect to $x$, we have
$\int {{a^{f\left( x \right)}}f’\left( x \right)dx} = \int {d\left[ {\frac{{{a^{f\left( x \right)}}}}{{\ln a}} + c} \right]}$

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.
$\int {{a^{f\left( x \right)}}f’\left( x \right)dx} = \frac{{{a^{f\left( x \right)}}}}{{\ln a}} + c$

Example: Evaluate the integral $\int {{5^{\sin x}}\cos xdx}$ with respect to $x$

We have integral $I = \int {{5^{\sin x}}\cos xdx}$

Here $f\left( x \right) = \sin x$ implies that $f’\left( x \right) = \cos x$, so using formula, we have
$\int {{5^{\sin x}}\cos xdx}$

Using the integration formula $\int {{a^{f\left( x \right)}}f’\left( x \right)dx = } \frac{{{a^{f\left( x \right)}}}}{{\ln a}} + c$, we have
$\int {{5^{\sin x}}\cos xdx} = {5^{\sin x}} + c$