6C1MoreExamples

More Examples of Integrating Trig Functions

Example

math . \qquad \displaystyle{ \int{ \sin^2 (x) \cos^3 (x) } \; dx} math

1. Separate a single instance of the odd power (cos)

math \\ . \qquad \displaystyle{ \int{ \sin^2 (x) \cos^3 (x) }\,dx } \\. \\ . \qquad = \displaystyle{ \int{ \big( \sin^2 (x) \big) \, \big( \cos^2 (x) \big) \, \big( \cos (x) \big) }\;dx } math

2. Use cos 2 (A) = 1 - sin 2 (A) on the remainder

math . \qquad = \displaystyle{ \int{ \big( \sin^2 (x) \big) \, \big( 1 - \sin^2 (x) \big) \, \big( \cos (x) \big) } \; dx } math

3. Let u = sin x (step 1 was cos x) and use the substitution method

math \\ . \qquad \text{Let } \; u = \sin (x) \\. \\ . \qquad \Rightarrow \; \dfrac {du}{dx} = \cos (x) \\. \\ . \qquad \Rightarrow \; dx = \dfrac {du}{\cos (x)} math

Hence

math \\ . \qquad \displaystyle{ \int{ \sin^2 (x) \cos^3 (x) } \; dx} \\. \\ . \qquad = \displaystyle{ \int{ \big( u^2 \big) \, \big( 1-u^2 \big) \, \big( \cos (x) \big) \;} \dfrac {du}{\cos (x)}} \\ .\\ . \qquad = \displaystyle{ \int{ u^2 - u^4 } \; du } math . math \\ . \qquad = \dfrac{u^3}{3} - \dfrac{u^5}{5} + c \\. \\ . \qquad = \dfrac{\sin^3 (x)}{3} - \dfrac{\sin^5 (x) }{5} + c math

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