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

Now consider

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

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

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

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

We have integral

Here $f\left( x \right) = \sin x$ implies that $f'\left( x \right) = \cos x$, so using formula, we have

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