Question

How do you evaluate the integral $ \int{{{e}^{4x}}dx} $ ?

Answer 1

Hint: We have to find the indefinite integral of $ \int{{{e}^{4x}}dx} $ . The given integral is in the form of $ \int{{{e}^{mx}}dx} $ where the indefinite integral gives us $ \int{{{e}^{mx}}dx}=\dfrac{{{e}^{mx}}}{m}+c $ . We find the value of $ m $ from the relation of $ mx=4x $ . Then we place the value of $ m $ in the integral to find the solution of the integral.

Complete step by step answer:
We need to find the indefinite integral of the given function $ \int{{{e}^{4x}}dx} $ . This is an exponential form. We are finding the integral of an exponent form with indices being $ 4x $ .
It is of the form $ \int{{{e}^{mx}}dx} $ where $ m $ is a constant.
We know that the indefinite integral of the $ \int{{{e}^{mx}}dx} $ is $ \int{{{e}^{mx}}dx}=\dfrac{{{e}^{mx}}}{m}+c $ .
For our given case the value of $ m $ will be decided from the relation of $ mx=4x $ .
We get the value of $ m $ as $ mx=4x\Rightarrow m=4 $ .
We apply the value of $ a=5 $ in the integral form of $ \int{{{e}^{mx}}dx}=\dfrac{{{e}^{mx}}}{m}+c $ to get \[\int{{{e}^{4x}}dx}=\dfrac{{{e}^{4x}}}{4}+c\].
Therefore, the indefinite integral value of $ \int{{{e}^{4x}}dx} $ is \[\dfrac{{{e}^{4x}}}{4}+c\].

Note:
 We need to remember that in case of the integral formula of $ \int{{{e}^{mx}}dx}=\dfrac{{{e}^{mx}}}{m}+c $ , we have to follow a condition which tells us that the value of m can never be 0. In that case the integral becomes undefined. $ \int{dx}=\dfrac{{{e}^{0}}}{0}+c $ . The term becomes 1 and on integration we get value of $ x $ .