MATH105 April 2014
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Question 01 (j)

Evaluate $\int {\frac {x13}{x^{2}x6}}dx$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

Factorize the denominator and use integration by partial fraction.

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Solution

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Factor the denominator:
$x^{2}x6=(x3)(x+2)$
Use partial fraction:
${\begin{aligned}{\frac {x13}{x^{2}x6}}&={\frac {A}{x3}}+{\frac {B}{x+2}}\\&={\frac {Ax+2A+Bx3B}{(x3)(x+2)}}\\&={\frac {(A+B)x+(2A3B)}{(x3)(x+2)}}\\\end{aligned}}$
Comparing the coefficients we get:
${\begin{aligned}A+B&=1\\2A3B&=13\\\end{aligned}}$
Solve and get:
${\begin{aligned}A&=2\\B&=3\\\end{aligned}}$
Hence:
${\begin{aligned}\int {\frac {x13}{x^{2}x6}}dx&=\int {\frac {2}{x3}}+{\frac {3}{x+2}}dx\\&=2\ln x3+3\ln x+2+C\\\end{aligned}}$

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