Find the integral of ∫ 0 π 2 sin ⁡ ( 2 θ ) ⋅ sin ⁡ ( θ ) d θ I got I = ∫ 0 π 4 sin ⁡ ( 2 θ ) ⋅ ( sin ⁡ ( θ ) + cos ⁡ ( θ ) ) d θ But, I'm stuck here.

Question

Find the integral of      ∫    0                  π        2                                sin        ⁡        (        2        θ        )              ⋅    sin    ⁡    (    θ    )    d    θ  I got  I  =      ∫    0          π      4                  sin      ⁡      (      2      θ      )        ⋅    (    sin    ⁡    (    θ    )    +    cos    ⁡    (    θ    )    )    d    θ  But, I&#039;m stuck here.

britspears523jp · Accepted Answer

If you are interested in knowing how to do this without using the Beta function, try the following steps. But I&#039;m not going to write it out in full because it would take too long.Call the integral I. First do integration by parts, and we find that  I  =      ∫    0                  π        2                        cos      ⁡      2      θ      cos      ⁡      θ              sin      ⁡      2      θ        d  θNow add this form of I to the original form and get  2  I  =      ∫    0                  π        2                        cos      ⁡      θ              sin      ⁡      2      θ        d  θhence  4  I  =      ∫    0                  π        2                        sin      ⁡      2      θ              sin      ⁡      θ        d  θNow substitute   t  =      tan    ⁡    θ   and you end up with a well known integral, featured many times on MSE, requiring a rather tedious partial fraction decomposition, but you get there in the end...I hope this is sufficient.

Abram Boyd · Accepted Answer

The integral  I  =      ∫          0              π              /            2            sin    ⁡    (    2    θ    )      sin  ⁡  θ    d  θis evaluated by making use of the Beta function. This is seen as follows.                    I                            =                  ∫                      0                                π                          /                        2                                    sin          ⁡          (          2          θ          )                        sin        ⁡        θ                d        θ                                          =                  2                                  ∫                      0                                π                          /                        2                                    sin                      3                          /                        2                          ⁡        (        θ        )                          cos                      1                          /                        2                          ⁡        (        θ        )                d        θ                                          =                  2                ⋅                  1          2                        B                  (                      3            4                    ,                      1            4                    )                                                  =                              Γ                          (                              1                4                            )                                    Γ                          (                              3                4                            )                                            4                                      2                                      =                  π          4                .

Find the integral of <msubsup> ∫ 0 <mrow class="MJX-TeXAtom-OR

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