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
\[\begin{gathered} \int {\left( {{7^x} + 3{x^2}} \right)dx} = {7^x}\frac{1}{{\ln 7}} + 3\frac{{{x^3}}}{3} + c \\ \Rightarrow \int {\left( {{7^x} + 3{x^2}} \right)dx} = {7^x}\frac{1}{{\ln 7}} + {x^3} + c \\ \end{gathered} \]