$\int \dfrac{dx}{(x+1)(x+2)}$ equals
A$\log\left|\dfrac{x+2}{x+1}\right|+C$
B$\log\left|\dfrac{x+1}{x+2}\right|+C$
C$\log|(x+1)(x+2)|+C$
D$\dfrac{1}{(x+1)(x+2)}+C$
Answer & Solution
Correct answer: B. $\log\left|\dfrac{x+1}{x+2}\right|+C$
1. Partial fractions: $\dfrac{1}{(x+1)(x+2)}=\dfrac{A}{x+1}+\dfrac{B}{x+2}$ gives $A=1,\ B=-1$.
2. So integrand $=\dfrac{1}{x+1}-\dfrac{1}{x+2}$.
3. Integrate: $\log|x+1|-\log|x+2|$.
4. Combine: $\log\left|\dfrac{x+1}{x+2}\right|+C$. Option A inverts the ratio.
_Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.317_
Related questions
$\displaystyle\int_2^3 \dfrac{x\,dx}{x^2+1}$ equals$\displaystyle\int_0^{2a} f(x)\,dx$ equals $2\displaystyle\int_0^{a} f(x)\,dx$ precisely wIf $f(a+b-x)=f(x)$, then $\displaystyle\int_a^b x\,f(x)\,dx$ is equal to$\displaystyle\int \frac{dx}{e^{x}+e^{-x}}$ is equal toA rational function $\dfrac{P(x)}{Q(x)}$ is called proper when$\displaystyle\int_0^{\pi/4} \tan x\,dx$ equals$\displaystyle\int_0^{1} x e^{x^2}\,dx$ equalsThe value of $\displaystyle\int_0^{\pi/2} \log\!\left(\dfrac{4+3 in x}{4+3\cos x}\right)dx