Trigonometric integrals
A variety of integrals can be solved by using the appropriate trigonometric identity.1
Evaluating
\int \sin^mx \cos^n x \, dxWhere
π β β 0
- If
π = 2 π + 1 β« s i n π β‘ π₯ c o s 2 π + 1 β‘ π₯ π π₯ = β« s i n π β‘ π₯ ( c o s 2 β‘ π₯ ) π c o s β‘ π₯ π π₯ = β« s i n π β‘ π₯ ( 1 β s i n 2 β‘ π₯ ) π c o s β‘ π₯ π π₯
- If
π = 2 π + 1 β« s i n 2 π + 1 β‘ π₯ c o s π β‘ π₯ π π₯ = β« ( s i n 2 β‘ π₯ ) π s i n β‘ π₯ c o s β‘ π₯ π π₯ = β« ( 1 β c o s 2 β‘ π₯ ) π c o s β‘ π₯ s i n β‘ π₯ π π₯
- If
and π are even, use the identities π s i n 2 β‘ π₯ = 1 2 ( 1 β c o s β‘ 2 π₯ ) c o s 2 β‘ π₯ = 1 2 ( 1 + c o s β‘ 2 π₯ ) s i n β‘ π₯ c o s β‘ π₯ = 1 2 s i n β‘ 2 π₯
Practice problems
- 2016. Calculus, Β§7.2, pp. 72β73 $