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
\[\int {{5^{\sin x}}\cos xdx} = {5^{\sin x}} + c\]