Worked example

∫ sin 3x cos x dx

Steps

1. sin 3x cos x
   A product of different functions — product-to-sum.
2. = ½[sin(3x + x) + sin(3x − x)]
   sin A cos B = ½[sin(A + B) + sin(A − B)]. · Product to sum: sin A cos B
3. = ½[sin 4x + sin 2x]
   Simplify the angles.
4. ∫ = ½[−cos 4x/4 − cos 2x/2] + C
   Integrate term by term.
5. = −cos 4x/8 − cos 2x/4 + C
   Collect; check by differentiating.

Answer line above uses the sign convention: ∫sin(kx)dx = −cos(kx)/k.

Identities used

• Product to sum: sin A cos B — A product of sine and cosine that you must integrate, sum, or compare at two frequencies. In calculus this is the standard opening move on ∫sin 3x cos x dx.

Check it yourself

Open the verifier with the main identity pre-filled — substituting your own numbers into a step is the fastest way to find out which line you did not follow.

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