Misc 37 - Prove that definite integral sin3 x dx = 2/3 - Miscellaneous

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Misc 37 Prove that ﷐0﷮﷐𝜋﷮2﷯﷮﷐﷐sin﷮3﷯﷮𝑥﷯﷯ 𝑑𝑥=﷐2﷮3﷯ Solving L.H.S ﷐0﷮﷐𝜋﷮2﷯﷮﷐﷐sin﷮3﷯﷮𝑥﷯﷯ 𝑑𝑥 = ﷐0﷮﷐𝜋﷮2﷯﷮ ﷐﷐sin 𝑥 (sin﷮2﷯﷮𝑥)﷯﷯ 𝑑𝑥 = ﷐0﷮﷐𝜋﷮2﷯﷮﷐𝑠𝑖𝑛﷮𝑥 (1−﷐𝑐𝑜𝑠﷮2﷯𝑥)﷯﷯ 𝑑𝑥 = ﷐0﷮﷐𝜋﷮2﷯﷮﷐𝑠𝑖𝑛﷮𝑥 𝑑𝑥− ﷐0﷮﷐𝜋﷮2﷯﷮﷐sin﷮𝑥 ﷐𝑐𝑜𝑠﷮2﷯𝑥﷯ 𝑑𝑥﷯﷯﷯ L.H.S = ﷐𝐼﷮1﷯− ﷐𝐼﷮2﷯ = 1 − ﷐1﷮3﷯ = ﷐𝟐﷮𝟑﷯ = R.H.S Hence, proved.

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