First slide

Introduction to integration

Question

$\int \frac{2^{x} + 3^{x}}{5^{x}} dx$ is equal to

Easy

A

$\frac{2^{x}}{\log 2} + \frac{3^{x}}{\log 3} - \frac{5^{x}}{\log 5} + C$
B

$\frac{2 / 5}{{xlog}_{e} (2 / 5)} + \frac{3 / 5}{{xlog}_{e} (3 / 5)} + C$
C

$\frac{(2 / 5)^{x}}{\log_{e} (2 / 5)} + \frac{(3 / 5)^{x}}{\log_{e} (3 / 5)} + C$
D

None of the above

Solution

$\int \frac{2^{x} + 3^{x}}{5^{x}} dx = \int (\frac{2^{x}}{5^{x}} + \frac{3^{x}}{5^{x}}) dx$
$\begin{array}{l} = \int [{(\frac{2}{5})}^{x} + {(\frac{3}{5})}^{x}] dx \\ = \frac{(2 / 5)^{x}}{\log_{e} 2 / 5} + \frac{(3 / 5)^{x}}{\log_{e} 3 / 5} + C \end{array}$

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