<p>I need help on taking complex integrals like int(sin^4x).</p>
<p>I now know how to do such problems, but whenever I approach these exercises, I keep on doing the wrong steps to get to the eventual answer. I can think of a million ways to split up the integral and such, but they always are wrong. Is there any way to expedite the process, or is the only way to get through these problems is to guess and check the right initial steps.</p>
<p>Yes there is a strategy,
(see page 3 and 4 of this document:
<a href=“http://www.stewartcalculus.com/data/CALCULUS%20Concepts%20and%20Contexts/upfiles/3c3-TrigonometIntegrals_Stu.pdf[/url]”>http://www.stewartcalculus.com/data/CALCULUS%20Concepts%20and%20Contexts/upfiles/3c3-TrigonometIntegrals_Stu.pdf</a>
but I always forget it. If you keep practicing with tons of integrals, however, you’ll get a better “feel” of how to solve them and make more correct guesses.</p>
<p>BTW, the problem you posted is solved on page 2 as example 4.</p>
<p>I keep this memorized:</p>
<p>sin/cos: change the ODD power function via sin^2 + cos^2 = 1</p>
<p>sec/tan: This is a bit trickier. the derivative of sec is almost always written sec * tan, rarely tan * sec. Also, the words “even, odd” fit nicely in that order because it is alphabetical.
So, I remember sec/tan -> even/odd.
If sec is even, change it.
If tan is odd, change it.</p>
<p>You’re trying to do a u-substitution method to solve the integral for these problems. In order to do this, you have to use a known trig identity. For the integral of sin^4x, try to do the substitution of sin^2x + cos^2x = 1, isolate sin^2x, plug it into the integral, and see if you can perform a successful u-sub</p>
<p>Use power-reduction.</p>
<p>sin^2 x = (1 - cos 2x)/2, squaring both sides gives sin^4 x. However this would involve another power-reduction identity (since you’re squaring cos 2x) so it gets messy.</p>
<p>According to Wikipedia,
sin^4 x = (3 - 4 cos 2x + cos 4x)/8</p>
<p>Replacing sin^4 x with (3 - 4 cos 2x + cos 4x)/8 makes it much easier to integrate.</p>