∫2(t^2 1)^2 (2)^2dt 2∫1(t^2 1)^2 4dt 2∫1(t^2 3)(t^2+1)dt ⇒ 2∫(14(t^2 3) 14(t^2+1))dt ⇒24∫1(t^2 3)dt 24∫1(t^2+1)dt ⇒12∫1t^2 3dt 12∫ ...

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Integration by Partial Fractions

∫ 2 t 2 -1 ...

Question

$\int \frac{2}{{(t^{2} - 1)}^{2} - {(2)}^{2}}$

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Solution

$\int \frac{2}{(t^{2} - 1)^{2} - (2)^{2}} d t$

$2 \int \frac{1}{(t^{2} - 1)^{2} - 4} d t$

$2 \int \frac{1}{(t^{2} - 3) (t^{2} + 1)} d t$

$\Rightarrow 2 \int (\frac{1}{4 (t^{2} - 3)} - \frac{1}{4 (t^{2} + 1)}) d t$

$\Rightarrow \frac{2}{4} \int \frac{1}{(t^{2} - 3)} d t - \frac{2}{4} \int \frac{1}{(t^{2} + 1)} d t$

$\Rightarrow \frac{1}{2} \int \frac{1}{t^{2} - 3} d t - \frac{1}{2} \int \frac{1}{t^{2} + 1} d t$

$\Rightarrow \frac{1}{4 \sqrt{3}} ln (\frac{t - \sqrt{3}}{t + \sqrt{3}}) - \frac{1}{2} {tan}^{- 1} t$

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