Integral of Constant to the Power of a Function

Integration of any constant power of some function is a general formula of exponential function and this formula need derivative of the given function and this formula has an importance in integral calculus.

The integration of any constant power of 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 , so the integral sign \int  {} and \frac{d}{{dx}} on the right side will cancel each other, 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 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