Question

stuck...
integrate (1/(x√x^2+4))dx

Answer 1

Does \[\LARGE x^2+4\] look like either of the following trig identities?\[\LARGE \sin^2 \theta + \cos ^2 \theta = 1 \\ \LARGE \tan^2 \theta + 1 = \sec^2 \theta\]

Answer 2

I would like to think \[\sin^2\theta + \cos^2 \theta = 1\] ?

Answer 3

Well, the reason we are looking at this is because we want to substitute something in that will allow the square root to go away.

So we need to pick the trig identity that has "something" PLUS a constant value, since we have (x^2+4) in the denominator we'd like to pull out of there.

So which one of those can do that or be rearranged to do that for us?

Answer 4

To be more clear, I'll show all three substitution types to look for: \[\LARGE -\sin^2 \theta +1 = \cos^2 \theta \\ \LARGE \tan^2 \theta+1 = \sec^2 \theta \\ \LARGE \sec^2 \theta - 1 = \tan^2 \theta\] See how the left side formulas parallel these possibilities? \[\LARGE -x^2+1 \\ \LARGE x^2+1 \\ \LARGE x^2-1\]

Answer 5

\[\tan^2\theta +1 = \sec^2 \theta \]
looks similar to x^2+4, but would I be able to convert my square root to \[x(x^2+4)(x^2+4) \] and bring the reciprocal out infront of the integration symbol?

Answer 6

\[\int\limits \frac{ dx }{ x \sqrt{x^2}-a^2 }= \frac{ 1 }{ a }\sec ^-1) \ln \frac{ x }{ a }+c \]

Answer 7

Not quite, how would you pull a reciprocal out of the integration symbol if it depends on x still?

Answer 8

Instead, multiply this equation by 4 on both sides to see: \[\LARGE 4\tan^2 \theta + 4 = 4 \sec^2 \theta\] Now we can see we should choose \[\LARGE x=2 \tan \theta\] since it will simplify this part of the expression: \[\large \sqrt{x^2+4}=\sqrt{(2 \tan \theta)^2+4}=\sqrt{4\tan^2 \theta+4}=2 \sec \theta\]

Answer 9

So for that last step if it seems obscure, I'm just doing this: \[\large \sqrt{4\tan^2 \theta +4}=\sqrt{4(\tan^2\theta+1)}=\sqrt{2^2 \sec^2 \theta}=2\sec \theta\]

Answer 10

Awesome, thanks Kainui