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Evaluate $\displaystyle \int \dfrac{2^{\displaystyle x}+3^{\displaystyle x}+4^{\displaystyle x}}{5^{\displaystyle x}} dx$

$2^{\displaystyle x}$, $3^{\displaystyle x}$, $4^{\displaystyle x}$ and $5^{\displaystyle x}$ are four exponential functions. The quotient of sum of $2^{\displaystyle x}$, $3^{\displaystyle x}$ and $4^{\displaystyle x}$ by $5^{\displaystyle x}$ is required to calculate by the integration.

Step: 1

The exponential function $5^{\displaystyle x}$ belongs to every term in the numerator. So, the entire function can be split into three functions mathematically.

$\displaystyle = \int \Bigg[\dfrac{2^{\displaystyle x}}{5^{\displaystyle x}} + \dfrac{3^{\displaystyle x}}{5^{\displaystyle x}} + \dfrac{4^{\displaystyle x}}{5^{\displaystyle x}}\Bigg] dx$

Apply integration to each function to obtain the integral of whole function.

$\displaystyle = \int \dfrac{2^{\displaystyle x}}{5^{\displaystyle x}} \, dx + \int \dfrac{3^{\displaystyle x}}{5^{\displaystyle x}} \, dx + \int \dfrac{4^{\displaystyle x}}{5^{\displaystyle x}} \, dx$

Step: 2

In every function, the base of exponential function in numerator is different from the base of exponential function in denominator but the exponents of both terms is same. So, each function can simplified by using quotient rule of exponents.

$\displaystyle = \int {\Big(\dfrac{2}{5}\Big)}^{\displaystyle x} \, dx + \int {\Big(\dfrac{3}{5}\Big)}^{\displaystyle x} \, dx + \int {\Big(\dfrac{4}{5}\Big)}^{\displaystyle x} \, dx$

Step: 3

The integral of each function be expressed by using the integral of a raised to the power x formula.

$\displaystyle = \dfrac{{\Big(\dfrac{2}{5}\Big)}^{\displaystyle x}}{\ln \Big(\dfrac{2}{5}\Big)} + \dfrac{{\Big(\dfrac{3}{5}\Big)}^{\displaystyle x}}{\ln \Big(\dfrac{3}{5}\Big)} + \dfrac{{\Big(\dfrac{4}{5}\Big)}^{\displaystyle x}}{\ln \Big(\dfrac{4}{5}\Big)} + C$

It is the required solution for this integral calculus problem.

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