Integration of Constant Power X

Integration of any constant of power x is another important integral calculus; this integral belongs to exponential formulas category and is one of the simplest formula 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 not must be not 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 , so the integral sign \int {} and \frac{d}{{dx}} on the right side will cancel each other, 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 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}