Question
Evaluate : $\quad \int \frac{1}{3-10 x-25 x^2} \cdot d x$
✓
Answer
$
\begin{aligned}
& I=\int \frac{1}{25\left(\frac{3}{25}-\frac{10}{25} x-x^2\right)} \cdot d x \\
& =\int \frac{1}{25\left[\frac{3}{25}-\left(x^2+\frac{2}{5} x\right)\right]} \cdot d x \\
& \because\left\{\left(\frac{1}{2} \text { coefficient of } x\right)^2\right. \\
& \left.=\left(\frac{1}{2}\left(\frac{2}{5}\right)\right)^2=\left(\frac{1}{5}\right)^2=\frac{1}{25}\right\} \\
& =\frac{1}{25} \cdot \int \frac{1}{\frac{3}{25}-\left(x^2-\frac{2}{5} x+\frac{1}{25}-\frac{1}{25}\right)} \cdot d x \\
& =\frac{1}{25} \cdot \int \frac{1}{\frac{3}{25}-\left(x^2-\frac{2}{5} x+\frac{1}{25}\right)+\frac{1}{25}} \cdot d x \\
& =\frac{1}{25} \cdot \int \frac{1}{\frac{4}{25}-\left(x^2-\frac{2}{5} x+\frac{1}{25}\right)} \cdot d x \\
& =\frac{1}{25} \cdot \int \frac{1}{\left(\frac{2}{5}\right)^2-\left(x-\frac{1}{5}\right)^2} \cdot d x \\
& \because \quad \int \frac{1}{a^2-x^2} \cdot d x=\frac{1}{2 a} \log \left(\frac{a+x}{a-x}\right)+c \\
& I=\frac{1}{25} \cdot \frac{1}{2\left(\frac{2}{5}\right)} \cdot \log \left(\frac{\frac{2}{5}+\left(x-\frac{1}{5}\right)}{\frac{2}{5}-\left(x-\frac{1}{5}\right)}\right)+c \\
& =\frac{1}{5} \cdot \log \left(\frac{1+5 x}{3-5 x}\right)+c \\
&
\end{aligned}
$
\begin{aligned}
& I=\int \frac{1}{25\left(\frac{3}{25}-\frac{10}{25} x-x^2\right)} \cdot d x \\
& =\int \frac{1}{25\left[\frac{3}{25}-\left(x^2+\frac{2}{5} x\right)\right]} \cdot d x \\
& \because\left\{\left(\frac{1}{2} \text { coefficient of } x\right)^2\right. \\
& \left.=\left(\frac{1}{2}\left(\frac{2}{5}\right)\right)^2=\left(\frac{1}{5}\right)^2=\frac{1}{25}\right\} \\
& =\frac{1}{25} \cdot \int \frac{1}{\frac{3}{25}-\left(x^2-\frac{2}{5} x+\frac{1}{25}-\frac{1}{25}\right)} \cdot d x \\
& =\frac{1}{25} \cdot \int \frac{1}{\frac{3}{25}-\left(x^2-\frac{2}{5} x+\frac{1}{25}\right)+\frac{1}{25}} \cdot d x \\
& =\frac{1}{25} \cdot \int \frac{1}{\frac{4}{25}-\left(x^2-\frac{2}{5} x+\frac{1}{25}\right)} \cdot d x \\
& =\frac{1}{25} \cdot \int \frac{1}{\left(\frac{2}{5}\right)^2-\left(x-\frac{1}{5}\right)^2} \cdot d x \\
& \because \quad \int \frac{1}{a^2-x^2} \cdot d x=\frac{1}{2 a} \log \left(\frac{a+x}{a-x}\right)+c \\
& I=\frac{1}{25} \cdot \frac{1}{2\left(\frac{2}{5}\right)} \cdot \log \left(\frac{\frac{2}{5}+\left(x-\frac{1}{5}\right)}{\frac{2}{5}-\left(x-\frac{1}{5}\right)}\right)+c \\
& =\frac{1}{5} \cdot \log \left(\frac{1+5 x}{3-5 x}\right)+c \\
&
\end{aligned}
$
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