Evaluate the following definite integral
$integral subscript 0 superscript straight pi over 4 end superscript space sin cubed space 2 straight t space cos space 2 straight t space dt$

Solution

Let I = $integral subscript 0 superscript straight pi over 4 end superscript space sin cubed space 2 straight t space cos space 2 straight t space dt$
Let $straight I subscript 1 space equals space integral space sin cubed space 2 straight t space cos space 2 straight t space dt$
Put sin 2t = y; $therefore space space 2 space cos space 2 dt space dt space equals space dy space space rightwards double arrow space space space space cos space 2 straight t space dt space equals space 1 half dy$
$therefore$ $straight I subscript 1 space equals space 1 half integral straight y cubed dy space equals space 1 half cross times straight y to the power of 4 over 4 space equals space 1 over 8 sin to the power of 4 2 straight t space equals space straight F left parenthesis straight t right parenthesis comma space say.$
By second fundamental theorem,
$straight I space equals space straight F open parentheses straight pi over 4 close parentheses minus straight F left parenthesis 0 right parenthesis space equals space 1 over 8 sin to the power of 4 straight pi over 2 minus 1 over 8 sin to the power of 4 0 space equals space 1 over 8 left parenthesis 1 right parenthesis to the power of 4 minus 1 over 8 left parenthesis 0 right parenthesis to the power of 4 space equals space 1 over 8 minus 0 space equals space 1 over 8$

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Integrals

Evaluate the following definite integral
$integral subscript 0 superscript straight pi over 4 end superscript space sin cubed space 2 straight t space cos space 2 straight t space dt$

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