Science:Math Exam Resources/Courses/MATH101/April 2017/Question 02 (b)

MATH101 April 2017

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Question 02 (b)
Which of the following substitutions is most helpful in evaluating the integral $\int _{2}^{4}{\frac {dx}{x^{2}{\sqrt {x^{2}+2x+10}}}}$? F: $x={\sqrt {10}}\tan u$ G: $u=x^{2}+2x+10$ H: $u=x^{2}$ J: $x=3\sec u-1$ K: $x=3\tan u-1$ L: $x={\sqrt {10}}\sec u$

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
Use the identity $\sec ^{2}u=\tan ^{2}u+1$ to simplify the inside of the square root.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We want an expression that simplifies the one on the square root. Note that $x^{2}+2x+10=(x+1)^{2}+3^{2}$, thus one of the expressions with a -1 on it will help us get rid of terms. The standard trigonometric identity with $\tan$ and $\sec$ is $\tan {^{2}}x+1=\sec {^{2}}x$. Note that if we substitute $x=3\tan u-1$ the expression on the inside of the square root becomes $(3\tan u-1+1)^{2}+3^{2}=3^{2}(\tan {^{2}}u+1)=(3\sec u)^{2}$ and this greatly simplifies the expression on the square root. Answer: $\color {blue}K:x=3\tan u-1$