# How do you integrate f(x)=5^x/2^x using the quotient rule?

Sep 25, 2016

$\int f \left(x\right) \mathrm{dx} = {5}^{x} / \left({2}^{x} \ln \left(\frac{5}{2}\right)\right) + C .$

#### Explanation:

In the first instance, there is no Rule called Quotient Rule for

Integration !

However, the fun. $f \left(x\right) = {5}^{x} / {2}^{x}$, can easily be integrated, using the

following Standard Integral :-

$\int {a}^{x} \mathrm{dx} = {a}^{x} / \ln a + C , a \in {\mathbb{R}}^{+} - \left\{1\right\} , x \in \mathbb{R}$.

Since, #5^x/2^x=(5/2)^x, we have,

$\int f \left(x\right) \mathrm{dx} = \int {\left(\frac{5}{2}\right)}^{x} \mathrm{dx} = {\left(\frac{5}{2}\right)}^{x} / \ln \left(\frac{5}{2}\right) = {5}^{x} / \left({2}^{x} \ln \left(\frac{5}{2}\right)\right) + C .$