Calculus II For Dummies
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You can integrate any function of the form sinm x cosn x when m is odd, for any real value of n. For this procedure, keep in mind the handy trig identity sin2 x + cos2 x = 1. For example, here’s how you integrate

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  1. Peel off a sin x and place it next to the dx:

    image1.png
  2. Apply the trig identity sin2 x = 1 – cos2 x to express the rest of the sines in the function as cosines:

    image2.png
  3. Use the variable substitution u = cos x and du = –sin x dx:

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Now that you have the function in terms of powers of u, the worst is over. You can expand the function out, turning it into a polynomial. This is just algebra:

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To continue, use the Sum Rule and Constant Multiple Rule to separate this into four integrals. Don’t forget to distribute that minus sign to all four integrals!

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At this point, you can evaluate each integral separately using the Power Rule:

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Finally, use u = cos x to reverse the variable substitution:

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Notice that when you substitute back in terms of x, the power goes next to the cos rather than next to the x, because you’re raising the entire function cos x to a power.

Similarly, you integrate any function of the form sinm x cosn x when n is odd, for any real value of m. These steps are practically the same as those in the previous example. For instance, here’s how you integrate sin–4 x cos9 x:

  1. Peel off a cos x and place it next to the dx:

    image8.png
  2. Apply the trig identity cos2 x = 1 – sin2 x to express the rest of the cosines in the function as sines:

    image9.png
  3. Use the variable substitution u = sin x and du = cos x dx:

    image10.png

At this point, you can distribute the function and express it as a sum of powers of u.

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Mark Zegarelli, a math tutor and writer with 25 years of professional experience, delights in making technical information crystal clear — and fun — for average readers. He is the author of Logic For Dummies and Basic Math & Pre-Algebra For Dummies.

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