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