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

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, the integral sign $\int {}$ and $\frac{d}{{dx}}$ on the right side will cancel each other out, 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 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