First slide

Methods of integration

Question

$\int (\frac{10 x^{9} + 10^{x} \log_{e} 10}{x^{10} + 10^{x}}) dx$ is equal to

Moderate

A

$10^{x} + x^{10} + C$
B

$10^{x} - x^{10} + C$
C

${(10^{x} - x^{10})}^{- 1} + C$
D

$\log |10^{x} + x^{10}| + C$

Solution

$\int (\frac{10 x^{9} + 10^{x} \log_{e} 10}{x^{10} + 10^{x}}) dx$
Let $x^{10} + 10^{x} = t \Rightarrow (10 x^{9} + 10^{x} \log_{e} 10) dx = dt$
$\begin{aligned} \Rightarrow dx = \frac{dt}{10 x^{9} + 10^{x} \log_{e} 10} \\ ∴ \int (\frac{10 x^{9} + 10^{x} \log_{e} 10}{x^{10} + 10^{x}}) dx = \int \frac{10 x^{9} + 10^{x} \log_{e} 10}{t} \times \frac{dt}{10 x^{2} + 10^{x} \log_{e} 10} \\ \begin{array}{l} = \int \frac{dt}{t} = \log | t | + C \\ = \log |10^{x} + x^{10}| + C \end{array} \end{aligned}$

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