Asked by Erica
How do you take the integral of dy/(y^2 + 4)?
Answers
Answered by
Steve
use the good old trig substitutions. When I first encountered them, they made no sense to me, but after putting the book away for a few weeks, I took another look, and it was obvious how useful they were.
Let y = 2 tan u
dy = 2 sec<sup>2</sup>u
Then
y<sup>2</sup> = 4 tan<sup>2</sup>u
y<sup>2</sup> + 4 = 4tan<sup>2</sup>u + 4 = 4sec<sup>2</sup>u
Now we have dy/4sec<sup>2</sup>u = 2sec<sup>2</sup>u/4sec<sup>2</sup>u = 1/2
So, integrating, we get u/2
But, since y = 2tan u, u = tan<sup>-1</sup>(y/2)
so, we end up with 1/2 tan<sup>-1</sup>(y/2) + C
Let y = 2 tan u
dy = 2 sec<sup>2</sup>u
Then
y<sup>2</sup> = 4 tan<sup>2</sup>u
y<sup>2</sup> + 4 = 4tan<sup>2</sup>u + 4 = 4sec<sup>2</sup>u
Now we have dy/4sec<sup>2</sup>u = 2sec<sup>2</sup>u/4sec<sup>2</sup>u = 1/2
So, integrating, we get u/2
But, since y = 2tan u, u = tan<sup>-1</sup>(y/2)
so, we end up with 1/2 tan<sup>-1</sup>(y/2) + C
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