# Integration of Constant of Power X

The integration of any constant of power $x$ is important and belongs to the exponential formulae. It is one of the simplest formulas of integration.

The integration of constant of power x is of the form
$\int {{a^x}dx = } \frac{1}{{\ln a}}{a^x} + c,\,\,\,a > 0,\,\,\,a \ne 1$

Where $a$ is any constant and must not be equal to zero.

Now consider
$\frac{d}{{dx}}\left[ {\frac{1}{{\ln a}}{a^x} + c} \right] = \frac{1}{{\ln a}}\frac{d}{{dx}}{a^x} + \frac{d}{{dx}}c$

Using the derivative formula $\frac{d}{{dx}}{a^x} = {a^x}\ln a$, we have
$\begin{gathered} \frac{d}{{dx}}\left[ {\frac{1}{{\ln a}}{a^x} + c} \right] = \frac{1}{{\ln a}}{a^x}\ln a + 0 \\ \Rightarrow \frac{d}{{dx}}\left[ {\frac{1}{{\ln a}}{a^x} + c} \right] = {a^x} \\ \Rightarrow {a^x} = \frac{d}{{dx}}\left[ {\frac{1}{{\ln a}}{a^x} + c} \right] \\ \Rightarrow {a^x}dx = d\left[ {\frac{1}{{\ln a}}{a^x} + c} \right]\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right) \\ \end{gathered}$

Integrating both sides of equation (i) with respect to $x$, we have
$\int {{a^x}dx} = \int {d\left[ {\frac{1}{{\ln a}}{a^x} + c} \right]}$

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.
$\int {{a^x}dx = } \frac{1}{{\ln a}}{a^x} + c$

Example: Evaluate the integral $\int {\left( {{7^x} + 3{x^2}} \right)dx}$ with respect to $x$

We have integral $I = \int {\left( {{7^x} + 3{x^2}} \right)dx}$
$\int {\left( {{7^x} + 3{x^2}} \right)dx} = \int {{7^x}dx + 3\int {{x^2}dx} }$

Using the integral formula $\frac{d}{{dx}}{a^x} = {a^x}\ln a$, we have
$\begin{gathered} \int {\left( {{7^x} + 3{x^2}} \right)dx} = {7^x}\frac{1}{{\ln 7}} + 3\frac{{{x^3}}}{3} + c \\ \Rightarrow \int {\left( {{7^x} + 3{x^2}} \right)dx} = {7^x}\frac{1}{{\ln 7}} + {x^3} + c \\ \end{gathered}$