Evaluate the following definite integrals: _ 1 ^ 2 x (x+1)(x+2) d… - Vidyadip

Question

Evaluate the following definite integrals:
$\int\limits_{1}^{2}\frac{\text{x}}{(\text{x}+1)(\text{x}+2)}\text{ dx}$

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Answer

Let $\text{I}=\int_{1}^\limits{2}\frac{\text{x}}{(\text{x}+1)(\text{x}+2)}\text{ dx}$ Then,
$\text{I}=\int_{1}^\limits{2}\Big(\frac{-1}{(\text{x}+1)}+\frac{2}{(\text{x}+2)}\Big)\text{dx}$
$\Rightarrow\text{I}=-\int_{1}^\limits{2}\frac{1}{(\text{x}+1)}\text{ dx}+2\int_{1}^\limits{2}\frac{1}{(\text{x}+2)}\text{ dx}$
$\Rightarrow\text{I}=\big[-\log(\text{x}+1)+2\log(\text{x}+2)\big]^2_1$
$\Rightarrow\text{I}=-\log3+2\log4+\log2-2\log3$
$\Rightarrow\text{I}=5\log2-3\log3$
$\Rightarrow\text{I}=\log2^5-\log3^3$
$\Rightarrow\text{I}=\log\frac{32}{27}$

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