Integration techniques MOC

Trigonometric integrals

A variety of integrals can be solved by using the appropriate trigonometric identity.¹

Evaluating

\int s i n 𝑚 𝑥 c o s 𝑛 𝑥 𝑑 𝑥

Where $𝑘 \in ℕ 0$

If $𝑛 = 2 𝑘 + 1$

\int s i n 𝑚 𝑥 c o s 2 𝑘 + 1 𝑥 𝑑 𝑥 = \int s i n 𝑚 𝑥 (c o s 2 𝑥) 𝑘 c o s 𝑥 𝑑 𝑥 = \int s i n 𝑚 𝑥 (1 - s i n 2 𝑥) 𝑘 c o s 𝑥 𝑑 𝑥

If $𝑚 = 2 𝑘 + 1$

\int s i n 2 𝑘 + 1 𝑥 c o s 𝑛 𝑥 𝑑 𝑥 = \int (s i n 2 𝑥) 𝑘 s i n 𝑥 c o s 𝑥 𝑑 𝑥 = \int (1 - c o s 2 𝑥) 𝑘 c o s 𝑥 s i n 𝑥 𝑑 𝑥

If $𝑛$ and $𝑚$ are even, use the identities

s i n 2 𝑥 = 12 (1 - c o s 2 𝑥) c o s 2 𝑥 = 12 (1 + c o s 2 𝑥) s i n 𝑥 c o s 𝑥 = 12 s i n 2 𝑥

Practice problems

2016. Calculus, §7.2, pp. 72–73 $

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2016. Calculus, §7.2 ↩