Techniques of Integration: Integration by Parts, Products of Powers of Trigonometric Functions Use integration by parts to integrate functions. Integrate products of powers of trigonometric functions. Evaluate ∫sin4θcos2θdθ

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Techniques of Integration: Integration by Parts, Products of Powers of Trigonometric Functions  Use integration by parts to integrate functions.  Integrate products of powers of trigonometric functions.  Evaluate  ∫sin4θcos2θdθ

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∫sin4θcos2θdθ On simplificaton, we get ∫sin4⁡θcos2⁡θdθ=∫(sin2⁡θ)2cos2⁡θdθ  =∫sin2θ(sin2θcos2θ)dθ =∫sin2θ(sin⁡θcos⁡θ)dθ =∫{(1−cos⁡2θ2}){(sin⁡2θ2})2dθ{[∵cos⁡2x=1−2sin2x &amp;amp; sin⁡2x=2sin⁡xcos⁡x}] =∫{(1−cos⁡2θ2}){(sin22θ4})dθ =18∫{(1−cos⁡2θ})(sin22θ)dθ =18∫{(sin22θ−cos⁡2θsin22θ})dθ =18∫{(1−cos⁡4θ2−cos⁡2θsin22θ})dθ =18∫{(1−cos⁡4θ−2cos⁡2θsin22θ2})dθ =116∫{(1−cos⁡4θ−2cos⁡2θsin22θ})dθ =116{[∫1dθ−∫cos⁡4θdθ−∫2cos⁡2θsin22θdθ}] Step 2 Now put, sin⁡2θ=t ⇒2cos⁡2θ=dtdθ ⇒2cos⁡2θdθ=dt Therefore,

Techniques of Integration: Integration by Parts, Products of Powers of

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