Analyze Circuits with Two Independent Sources Using Superposition
Analyze Noninverting Op Amp Circuits
How to Integrate Tangent/Secant Problems with an Even, Positive Power of Secant

How to Integrate Sine/Cosine Problems with an Odd, Positive Power of Sine

Here’s how you integrate a trig integral that contains sines and cosines where the power of sine is odd and positive. You lop off one sine factor and put it to the right of the rest of the expression, convert the remaining (even) sine factors to cosines with the Pythagorean identity, and then integrate with the substitution method where u = cos(x).

Remember that the Pythagorean identity tells you that, for any angle x,

image0.png

And thus,

image1.png
  1. Lop off one sine factor and move it to the right.

    image2.png
  2. Convert the remaining (even) sines to cosines by using the Pythagorean identity and simplify.

    image3.png
  3. Integrate with substitution, where u = cos(x).

    image4.png

You can save a little time in all substitution problems by just solving for du—as is done immediately above — and not bothering to solve for dx. You then tweak the integral so that it contains the thing du equals (–sin(x)dx in this problem). The integral contains a sin(x)dx, so you multiply it by –1 to turn it into –sin(x)dx and then compensate for that –1 by multiplying the whole integral by –1. This is a wash because –1 times –1 equals 1. This may not sound like much of a shortcut, but it’s a good time saver once you get used to it.

So tweak your integral:

image5.png

Now substitute and solve by the reverse power rule:

image6.png

It’s a walk in the park.

  • Add a Comment
  • Print
  • Share
blog comments powered by Disqus
Analyze a First-Order RL Circuit Using Laplace Methods
Find the Power and Energy of a Capacitor
How to Use Trig Substitution to Integrate radicals of the sine form
How to Perform Complex Processing with Op Amps
How to Solve Differential Equations Using Op Amps
Advertisement

Inside Dummies.com