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Asked by dk20062000shaurya | 13th May, 2020, 11:30: AM

Expert Answer:

To find the integration of sin3theta cos-1theta w.r.t theta between the limits 0 to n/2
L e t space I equals integral subscript 0 superscript n divided by 2 end superscript sin cubed theta space cos to the power of negative 1 end exponent theta space d theta equals integral subscript 0 superscript n divided by 2 end superscript fraction numerator sin squared theta space sin theta over denominator cos theta end fraction space d theta equals integral subscript 0 superscript n divided by 2 end superscript fraction numerator open parentheses 1 minus cos squared theta close parentheses space sin theta over denominator cos theta end fraction space d theta T a k e space cos theta equals t space rightwards double arrow space sin theta space d theta space equals space minus d t rightwards double arrow I equals integral subscript 0 superscript n divided by 2 end superscript fraction numerator open parentheses 1 minus t squared close parentheses over denominator t end fraction d t equals integral subscript 0 superscript n divided by 2 end superscript open parentheses 1 over t minus t close parentheses d t space space space space space equals open square brackets log t minus t squared over 2 close square brackets subscript 0 superscript n divided by 2 end superscript space space space space space equals open square brackets log open parentheses cos theta close parentheses minus open parentheses cos theta close parentheses squared over 2 close square brackets subscript 0 superscript n divided by 2 end superscript space space space space space equals log open parentheses cos 0 close parentheses minus open parentheses cos 0 close parentheses squared over 2 minus log open parentheses cos n over 2 close parentheses plus open parentheses cos begin display style n over 2 end style close parentheses squared over 2 rightwards double arrow I equals negative 1 half minus log open parentheses cos n over 2 close parentheses plus open parentheses cos begin display style n over 2 end style close parentheses squared over 2

Answered by Renu Varma | 17th May, 2020, 11:21: AM