How to evaluate the Trigonometric Integrals : ∫ sin^2(1/3 θ) dθ , the upper limit is 2π and the lower limit is 0 ?

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Answer 1 · 2015-10-14T00:47:13

Use the power reducing formula.

From #cos(2t) = 1-2sin^2t#, we get #sin^2t = 1/2(1-cos(2t))#.

#int sin^2t dt = 1/2 int (1-cos(2t)) dt#, which is straightforward to integrat by substitution.

In this question, #t = 1/3theta#, so we get

#int sin^2(1/3theta) d theta = 1/2 int (1-cos(2/3theta)) d theta#.

Integrate #1 d theta#, then #cos(2/3theta) d theta# by subsitution.

Note

We didn't need it here, but. also from #cos(2t) = 2cos^2t-1#, we get the power reduction for cosine.

#cos^2t = 1/2(cos(2t)-1)#