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Integrals

Question

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Evaluate: $integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sin squared straight x over denominator sinx plus cosx end fraction dx.$

Solution

Let space straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sin squared straight x over denominator sinx plus cosx end fraction dx space space space... left parenthesis straight i right parenthesis rightwards double arrow space space space straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sin squared open parentheses begin display style straight pi over 2 end style minus straight x close parentheses over denominator sin open parentheses begin display style straight pi over 2 end style minus straight x close parentheses plus cos open parentheses begin display style straight pi over 2 end style minus straight x close parentheses end fraction dx space left curly bracket Using space Property space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis dx space equals integral subscript 0 superscript straight a straight f left parenthesis straight a minus straight x right parenthesis dx right curly bracket rightwards double arrow space straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator cos squared straight x over denominator sinx plus cosx end fraction dx.... left parenthesis ii right parenthesis Adding space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma rightwards double arrow space 2 straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sin squared straight x plus cos squared straight x over denominator sinx plus cosx end fraction dx rightwards double arrow space 2 straight I space equals space integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator dx over denominator sinx plus cosx end fraction rightwards double arrow 2 straight I space equals space fraction numerator 1 over denominator square root of 2 end fraction integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator dx over denominator sinx. begin display style fraction numerator 1 over denominator square root of 2 end fraction end style plus cosx begin display style fraction numerator 1 over denominator square root of 2 end fraction end style end fraction rightwards double arrow 2 straight I space equals space fraction numerator 1 over denominator square root of 2 end fraction integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator dx over denominator sinx. space cos begin display style straight pi over 4 end style plus cosx. sin begin display style straight pi over 4 end style end fraction

rightwards double arrow space 2 straight I space equals fraction numerator 1 over denominator square root of 2 end fraction integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator dx over denominator sin open parentheses begin display style straight pi over 4 end style plus straight x close parentheses end fraction rightwards double arrow space 2 straight I space equals space fraction numerator 1 over denominator square root of 2 end fraction integral subscript 0 superscript straight pi divided by 2 end superscript cosec open parentheses straight pi over 4 plus straight x close parentheses dx rightwards double arrow space 2 straight I space equals fraction numerator 1 over denominator square root of 2 end fraction open square brackets In open vertical bar cosec open parentheses straight pi over 4 plus straight x close parentheses minus cot open parentheses straight pi over 4 plus straight x close parentheses close vertical bar close square brackets subscript 0 superscript straight pi divided by 2 end superscript rightwards double arrow space 2 straight I space equals space fraction numerator 1 over denominator square root of 2 end fraction open square brackets ln open vertical bar cosec open parentheses straight pi over 4 plus straight pi over 2 close parentheses minus cot open parentheses straight pi over 4 plus straight pi over 2 close parentheses close vertical bar minus ln open vertical bar cosec open parentheses straight pi over 4 plus 0 close parentheses minus cot open parentheses straight pi over 4 plus 0 close parentheses close vertical bar close square brackets rightwards double arrow space 2 straight I space equals space fraction numerator 1 over denominator square root of 2 end fraction open square brackets In open vertical bar square root of 2 minus left parenthesis negative 1 right parenthesis close vertical bar minus In open vertical bar square root of 2 minus 1 close vertical bar close square brackets rightwards double arrow straight I space equals space fraction numerator 1 over denominator 2 square root of 2 end fraction open square brackets In open vertical bar fraction numerator square root of 2 plus 1 over denominator square root of 2 minus 1 end fraction close vertical bar close square brackets

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