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Integrals — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIntegrals3 Marks
Question
Evaluate $\int \frac{5 x-2}{1+2 x+3 x^2} d x$
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Answer

Express numerator as $5 x-2=A \frac{d}{d x}\left(3 x^2+2 x+1\right)+B 5 x-2$
$A(6 x+$ 2) $+B$
Comparing coefficients:
$6 A=5 \Longrightarrow A=5 / 6$ and $2 A+B=-2$
$\Longrightarrow B=$ $-11 / 3$
$I=\frac{5}{6} \int \frac{6 x+2}{3 x^2+2 x+1} d x-\frac{11}{3} \int \frac{d x}{3\left(x^2+\frac{2}{3} x+\frac{1}{3}\right)}$
The first part is $\frac{5}{6} \ln \left|3 x^2+2 x+1\right|$.
For the second part, complete the square: $x^2+\frac{2}{3} x+\frac{1}{3}$
$=\left(x+\frac{1}{3}\right)^2+\left(\frac{\sqrt{2}}{3}\right)^2$
Using $\int \frac{d x}{x^2+a^2}=\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)$
$\frac{5}{6} \ln \left|3 x^2+2 x+1\right|-\frac{11}{3 \sqrt{2}} \tan ^{-1}\left(\frac{3 x+1}{\sqrt{2}}\right)+C$

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