Question

Evaluate:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator straight a space cosx plus straight b space sinx end fraction. space straight a comma space straight b space greater than space 0$

Solution

Let I = $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator acosx plus bsinx end fraction$
Put $tan space straight x over 2 space equals space straight t space space space or space straight x over 2 space equals space tan to the power of negative 1 end exponent space straight t space rightwards double arrow space space straight x space equals space 2 space tan to the power of negative 1 end exponent straight t space space space space rightwards double arrow space space space dx space equals space fraction numerator 2 over denominator 1 plus straight t squared end fraction dt$
When $straight x space equals space 0 comma space space straight t space equals space tan space 0 space equals space 0$
When $straight x space equals space straight pi over 2 comma space straight t space equals space tan straight pi over 4 space equals space 1$
Also $sinx space equals space fraction numerator 2 space tan space begin display style straight x over 2 end style over denominator 1 plus tan squared begin display style straight x over 2 end style end fraction space equals fraction numerator 2 straight t over denominator 1 minus straight t squared end fraction comma space space cosx space equals space fraction numerator 1 minus tan squared begin display style straight x over 2 end style over denominator 1 plus tan squared begin display style straight x over 2 end style end fraction space equals space fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction$
$therefore space space space space straight I space equals space integral subscript 0 superscript 1 fraction numerator begin display style fraction numerator 2 over denominator 1 plus straight t squared end fraction end style dt over denominator straight a open parentheses begin display style fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction end style close parentheses plus straight b open parentheses begin display style fraction numerator 2 straight t over denominator 1 plus straight t squared end fraction end style close parentheses end fraction space equals space integral subscript 0 superscript 1 fraction numerator 2 dt over denominator straight a left parenthesis 1 minus straight t squared right parenthesis plus 2 bt end fraction$
$equals space 2 over straight a integral subscript 0 superscript 1 fraction numerator dt over denominator 1 minus straight t squared plus begin display style fraction numerator 2 straight b over denominator straight a end fraction end style straight t end fraction straight t space equals space 2 over straight a integral subscript 0 superscript 1 fraction numerator dt over denominator 1 minus open parentheses straight t squared minus begin display style fraction numerator 2 straight b over denominator straight a end fraction end style straight t close parentheses end fraction equals space 2 over straight a integral subscript 0 superscript 1 fraction numerator dt over denominator open parentheses 1 plus begin display style straight b squared over straight a squared end style close parentheses minus open parentheses straight t squared minus begin display style fraction numerator 2 straight b over denominator straight a end fraction end style straight t plus begin display style straight b squared over straight a squared end style close parentheses end fraction space equals space 2 over straight a integral subscript 0 superscript 1 fraction numerator dt over denominator open parentheses begin display style fraction numerator square root of straight a squared plus straight b squared end root over denominator straight a end fraction end style close parentheses squared minus open parentheses straight t minus begin display style straight b over straight a end style close parentheses squared end fraction equals space 2 over straight a fraction numerator 1 over denominator begin display style fraction numerator 2 square root of straight a squared plus straight b squared end root over denominator straight a end fraction end style end fraction open square brackets fraction numerator open vertical bar log space begin display style fraction numerator square root of straight a squared plus straight b squared end root over denominator straight a end fraction end style plus open parentheses straight t minus begin display style straight b over straight a end style close parentheses close vertical bar over denominator begin display style fraction numerator square root of straight a squared plus straight b squared end root over denominator straight a end fraction end style minus open parentheses straight t minus begin display style straight b over straight a end style close parentheses end fraction close square brackets subscript 0 superscript 1$
$equals space fraction numerator 1 over denominator square root of straight a squared plus straight b squared end root end fraction open square brackets log open square brackets fraction numerator open parentheses begin display style fraction numerator square root of straight a squared plus straight b squared end root over denominator straight a end fraction end style plus 1 minus begin display style straight b over straight a end style close parentheses over denominator begin display style fraction numerator square root of straight a squared plus straight b squared end root over denominator straight a end fraction end style minus 1 plus begin display style straight b over straight a end style end fraction close square brackets minus space log space open parentheses fraction numerator begin display style fraction numerator square root of straight a squared plus straight b squared end root over denominator straight a end fraction minus straight b over straight a end style over denominator begin display style fraction numerator square root of straight a squared plus straight b squared end root over denominator straight a end fraction plus straight b over straight a end style end fraction close parentheses close square brackets space equals space fraction numerator 1 over denominator square root of straight a squared plus straight b squared end root end fraction open square brackets log space open parentheses fraction numerator square root of straight a squared plus straight b squared end root plus straight a minus straight b over denominator square root of straight a squared plus straight b squared end root minus straight a plus straight b end fraction close parentheses minus log space open parentheses fraction numerator square root of straight a squared plus straight b squared end root minus straight b over denominator square root of straight a squared plus straight b squared end root plus straight b end fraction close parentheses close square brackets$
$space equals space fraction numerator 1 over denominator square root of straight a squared plus straight b squared end root end fraction space log space open parentheses fraction numerator square root of straight a squared plus straight b squared end root plus straight a minus straight b over denominator square root of straight a squared plus straight b squared end root minus straight a plus straight b end fraction cross times fraction numerator square root of straight a squared plus straight b squared end root plus straight b over denominator square root of straight a squared plus straight b squared end root minus straight b end fraction close parentheses equals space fraction numerator 1 over denominator square root of straight a squared plus straight b squared end root end fraction log open parentheses fraction numerator straight a squared plus straight b squared plus ab minus straight b squared plus straight a square root of straight a squared plus straight b squared end root over denominator straight a squared plus straight b squared plus ab minus straight b squared minus straight a square root of straight a squared plus straight b squared end root end fraction close parentheses equals space fraction numerator 1 over denominator square root of straight a squared plus straight b squared end root end fraction log open parentheses fraction numerator straight a plus straight b plus square root of straight a squared plus straight b squared end root over denominator straight a plus straight b minus square root of straight a squared plus straight b squared end root end fraction close parentheses$

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Integrals

Evaluate:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator straight a space cosx plus straight b space sinx end fraction. space straight a comma space straight b space greater than space 0$

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