Evaluate the following:
$integral subscript 0 superscript straight pi fraction numerator 1 over denominator 5 plus 2 cosx end fraction dx$

Solution

Let I = $integral subscript 0 superscript straight pi fraction numerator 1 over denominator 5 plus 2 cosx end fraction dx$
Put $tan straight x over 2 space equals space straight t comma space space space or space space straight x over 2 space equals space tan to the power of negative 1 end exponent straight t comma space space space or space straight x space equals space 2 space tan to the power of negative 1 end exponent straight t space rightwards double arrow space space space space dx space equals space fraction numerator 2 over denominator 1 plus straight t squared end fraction dt$
When x = 0, t = tan 0 = 0
When $straight x equals straight pi comma space space straight t space equals tan straight pi over 2 rightwards arrow infinity$
Also $cosx space equals space fraction numerator 1 minus tan squared begin display style straight x over 2 end style over denominator 1 plus tan squared begin display style straight x over 2 end style end fraction space equals space fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction$
$therefore$ $straight I space equals space integral subscript 0 superscript infinity fraction numerator begin display style fraction numerator 2 over denominator 1 plus straight t squared end fraction end style dt over denominator 5 plus 2 open parentheses begin display style fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction end style close parentheses end fraction space equals space integral subscript 0 superscript infinity fraction numerator 2 over denominator 5 space left parenthesis 1 plus straight t squared right parenthesis space plus 2 space left parenthesis 1 minus straight t squared right parenthesis end fraction dt$
$equals space 2 space integral subscript 0 superscript infinity fraction numerator 1 over denominator 3 straight t squared plus 7 end fraction dt space equals space 2 over 3 integral subscript 0 superscript infinity fraction numerator 1 over denominator straight t squared plus begin display style 7 over 3 end style end fraction dt space equals space 2 over 3 integral subscript 0 superscript infinity fraction numerator 1 over denominator straight t squared plus open parentheses begin display style fraction numerator square root of 7 over denominator 3 end fraction end style close parentheses squared end fraction dt$
$equals space 2 over 3 fraction numerator 1 over denominator square root of begin display style 7 over 3 end style end root end fraction open square brackets tan to the power of negative 1 end exponent open parentheses fraction numerator straight t over denominator square root of begin display style 7 over 3 end style end root end fraction close parentheses close square brackets subscript 0 superscript infinity space equals space fraction numerator 2 over denominator square root of 21 end fraction left square bracket tan to the power of negative 1 end exponent infinity space minus space tan to the power of negative 1 end exponent 0 right square bracket space equals fraction numerator 2 over denominator square root of 21 end fraction open square brackets straight pi over 2 minus 0 close square brackets space equals space fraction numerator straight pi over denominator square root of 21 end fraction$

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Integrals

Evaluate the following:
$integral subscript 0 superscript straight pi fraction numerator 1 over denominator 5 plus 2 cosx end fraction dx$

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