Question
Download Solution PDFEvaluate \(\mathop \smallint \limits_1^2 \frac{{xdx}}{{\left( {x + 1} \right)\left( {x + 2} \right)}}\)
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDF\(Let\;I = \mathop \smallint \limits_1^2 \frac{{xdx}}{{\left( {x + 1} \right)\left( {x + 2} \right)}}\)
Using partial fraction, we get \(\frac{x}{{\left( {x + 1} \right)\left( {x + 2} \right)}} = \frac{{ - 1}}{{x + 1}} + \frac{2}{{x + 2}}\)
So, \(\smallint \frac{{x\;dx}}{{\left( {x + 1} \right)\left( {x + 2} \right)}} = - \log \left| {x + 1} \right| + 2\log \left| {x + 2} \right| = F\left( x \right)\)
I = F (2) – F(1)
= [- log3 + 2log 4] – [- log2 + 2 log3]
\( = - {\rm{\;}}3{\rm{\;log\;}}3 + {\rm{log\;}}2 + 2{\rm{\;log\;}}4 = {\rm{log}}\left( {\frac{{32}}{{27}}} \right)\)
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