$\int \frac{t}{t^{4} + t^{2} - 5} d t is equal to$

Moderate

A

$\frac{1}{\sqrt{21}} \log |\frac{2 t^{2} + 1 + \sqrt{21}}{2 t^{2} + 1 + \sqrt{21}}| + c$
B

$\frac{1}{\sqrt{21}} \log |\frac{2 t^{2} - 1 + \sqrt{21}}{2 t^{2} + 1 + \sqrt{21}}| + c$
C

$\frac{1}{2 \sqrt{21}} \log |\frac{2 t^{2} - 1 + \sqrt{21}}{2 t^{2} + 1 + \sqrt{21}}|$
D

$\frac{1}{2 \sqrt{21}} \log |\frac{2 t^{2} + 1 - \sqrt{21}}{2 t^{2} + 1 + \sqrt{21}}| + c$

Solution

$\int \frac{t}{t^{4} + t^{2} - 5} d t$
$t^{2} = z$
$\begin{array}{l} 2 t d t = d z \\ \int \frac{t}{{(t^{2})}^{2} + t^{2} - 5} d t = \int \frac{d z}{z^{2} + z - 5} \\ = \frac{1}{2} \int \frac{d z}{z^{2} + 2 . z \frac{1}{2} + {(\frac{1}{2})}^{2} - 5 - {(\frac{1}{2})}^{2}} \end{array}$
$Use \int \frac{1}{x^{2} - a^{2}} d x = \frac{1}{2 a} \log |\frac{x - a}{x + a}| + c$
$\begin{array}{l} = \frac{1}{2} \int \frac{d z}{{(z + \frac{1}{2})}^{2} - \frac{21}{4}} \\ = \frac{1}{2} \int \frac{d z}{{(z + \frac{1}{2})}^{2} - {(\frac{\sqrt{21}}{2})}^{2}} \\ = \frac{1}{2} \frac{1}{2 \frac{\sqrt{21}}{2}} \log |\frac{(z + \frac{1}{2}) - \frac{\sqrt{21}}{2}}{(z + \frac{1}{2}) + \frac{\sqrt{21}}{2}}| + c \\ = \frac{1}{2 \sqrt{21}} \log |\frac{2 z + 1 - \sqrt{21}}{2 z + 1 + \sqrt{21}}| + c \end{array}$

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