MATH101 April 2007
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Question 03 (c)

FullSolution Problem. Justify your answers and show all your work. Simplification of answers is not required.
Evaluate the following integral.
$\int {\frac {dx}{\sqrt {x^{2}+2x+5}}}$

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1

Try completing the square.

Hint 2

Try a trig substitution.

Hint 3

Try the trig substitution $x+1=2\tan \theta$

Hint 4

The integral of $\sec \theta$ should appear. This was one of the trickier integrals we did in the course. What was the trick we used?

Hint 5

To evaluate
$\int \sec \theta \,d\theta$
Try multiplying top and bottom by
${\frac {\sec \theta +\tan \theta }{\sec \theta +\tan \theta }}.$

Hint 6

The do the substitution $u=\sec \theta +\tan \theta$.

Hint 7

Don't forget to plug back to the original variables. You will need to draw a triangle and figure out what $\sec \theta$ is in terms of x.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
 If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
 If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

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First, we complete the square in our integral to get
 $\int {\frac {dx}{\sqrt {x^{2}+2x+5}}}=\int {\frac {dx}{\sqrt {x^{2}+2x+1+4}}}=\int {\frac {dx}{\sqrt {(x+1)^{2}+4}}}$
Now, let $x+1=2\tan \theta$ so that $dx=2\sec ^{2}\theta$. This gives
 $\int {\frac {dx}{\sqrt {(x+1)^{2}+4}}}=\int {\frac {2\sec ^{2}\theta d\theta }{\sqrt {4\tan ^{2}\theta +4}}}$
Using the trig identity $\tan ^{2}\theta +1=\sec ^{2}\theta$, we have
 $\int {\frac {2\sec ^{2}\theta d\theta }{\sqrt {4\tan ^{2}\theta +4}}}=\int {\frac {2\sec ^{2}\theta d\theta }{\sqrt {4\sec ^{2}\theta }}}=\int \sec \theta d\theta$
This last integral has a clever trick. Multiply top and bottom by
 ${\frac {\sec \theta +\tan \theta }{\sec \theta +\tan \theta }}$
to get
 ${\begin{aligned}\int \sec \theta d\theta &=\int \sec \theta \cdot {\frac {\sec \theta +\tan \theta }{\sec \theta +\tan \theta }}d\theta \\&=\int {\frac {\sec ^{2}\theta +\sec \theta \tan \theta }{\sec \theta +\tan \theta }}d\theta \\\end{aligned}}$
Let $u=\sec \theta +\tan \theta$ so $du=\sec ^{2}\theta +\sec \theta \tan \theta$ and so the above integral is
 ${\begin{aligned}\int {\frac {\sec ^{2}\theta +\sec \theta \tan \theta }{\sec \theta +\tan \theta }}d\theta &=\int {\frac {du}{u}}\\&=\ln u\\&=\ln \sec \theta +\tan \theta \end{aligned}}$
To get x back in the expression we need to draw our triangle.
(Here we used Pythagorean theorem to get the hypotenuse.) From the picture, we see that
 $\cos \theta ={\frac {2}{\sqrt {(x+1)^{2}+4}}}$
and so
 $\sec \theta ={\frac {\sqrt {(x+1)^{2}+4}}{2}}$
Thus,
 $\int {\frac {dx}{\sqrt {x^{2}+2x+5}}}=\ln \sec \theta +\tan \theta =\ln \left{\frac {\sqrt {(x+1)^{2}+4}}{2}}+{\frac {x+1}{2}}\right$
completing the question.

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