Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIntegrals1 Mark
Question
Find: $\int \frac{x^{2}}{\left(x^{2}+1\right)\left(x^{2}+4\right)} d x$
✓
Answer
Consider $\frac{x^{2}}{\left(x^{2}+1\right)\left(x^{2}+4\right)}$ and put $x^2 = y$
Then, $\frac{x^{2}}{\left(x^{2}+1\right)\left(x^{2}+4\right)}=\frac{y}{(y+1)(y+4)}$
Write, $\frac{y}{(y+1)(y+4)}=\frac{A}{y+1}+\frac{B}{y+4}$
So that, $y = A (y + 4) + B (y + 1)$
Comparing coefficients of $y$ and constant terms on both sides,
we get $A + B = 1$ and $4A + B = 0,$ which give
$A=-\frac{1}{3}$ and $B=\frac{4}{3}$
Thus, $\frac{x^{2}}{\left(x^{2}+1\right)\left(x^{2}+4\right)}=-\frac{1}{3\left(x^{2}+1\right)}+\frac{4}{3\left(x^{2}+4\right)}$
$= -\frac{1}{3} \tan ^{-1} x+\frac{4}{3} \times \frac{1}{2} \tan ^{-1} \frac{x}{2}+C$
$= -\frac{1}{3} \tan ^{-1} x+\frac{2}{3} \tan ^{-1} \frac{x}{2}+C$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.