First slide

Methods of integration

Question

$\int \frac{t^{4} d t}{1 - t^{10}} i s$

Moderate

A

$\frac{1}{10} \log |\frac{1 - t^{5}}{1 + t^{5}}| + c$
B

$\frac{1}{5} \log |\frac{1 + t^{5}}{1 - t^{5}}| + c$
C

$\frac{- 1}{10} \log |\frac{1 - t^{5}}{1 + t^{5}}| + c$
D

$\frac{1}{10} \log |\frac{1 + t^{5}}{1 - t^{5}}| + c$

Solution

$\int \frac{t^{4} d t}{1 - t^{10}} d t = \int \frac{t^{4} d t}{1 - {(t^{5})}^{2}}$
$t^{5} = u \Rightarrow 5 t^{4} d t = d u$
$Use \int \frac{1}{a^{2} - x^{2}} d x = \frac{1}{2 a} \log |\frac{a + x}{a - x}| + c = \frac{1}{5} \int \frac{d u}{1^{2} - u^{2}}$
$= \frac{1}{5} \cdot \frac{1}{2 (1)} \log |\frac{1 + u}{1 - u}| + c$
$= \frac{1}{10} \log |\frac{1 + t^{5}}{1 - t^{5}}| + c$

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