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, $\frac{x^{2}}{\left(x^{2}+1\right)\left(x^{2}+4\right)}=-\frac{1}{3\left(x^{2}+1\right)}+\frac{4}{3\left(x^{2}+4\right)}$
= $-\frac{1}{3} \tan ^{-1} x+\frac{4}{3} \times \frac{1}{2} \tan ^{-1} \frac{x}{2}+C$
= $-\frac{1}{3} \tan ^{-1} x+\frac{2}{3} \tan ^{-1} \frac{x}{2}+C$
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.
Explore more
Similar questions
What will be the differential equation of the curve $y=$ $A e^x+ B e^{-x}$ ?
→Classify the following as scalar and vector quantities:
Displacement.
Displacement.
In triangle ABC, which of the following is not true:
Evaluate: $\int\limits^{3}_{2} 3^{x} \text{d}x.$
→Represent graphically a displacement of 40 km, 30° east of north.
If the points $(1,2,3 \lambda),(1,0,1)$ and $(2,1,2)$ are collinear then find the value of $\lambda$.
→What happens to a function f(x) at x = a, if $\lim\limits_{{\text{x}}\rightarrow\text{a}}\text{f(x})=\text{f}(\text{a})?$
→Find the principal value of the following:
$\sin^{-1}\Big(\cos\frac{3\pi}{4}\Big)$
→$\sin^{-1}\Big(\cos\frac{3\pi}{4}\Big)$
Find: $\int \frac{3 x-2}{(x+1)^{2}(x+3)} d x$
→A relation $R$ is defined as $A R B \Leftrightarrow A$ is subset of $B$ is sets of set S . Is this relation R will anti symmetric?
→