Question

Evaluate $integral subscript 0 superscript 2 straight pi end superscript fraction numerator cosx over denominator square root of 4 plus 3 sinx end root end fraction dx.$

Solution

Let I = $integral subscript 0 superscript 2 straight pi end superscript fraction numerator cos space straight x over denominator square root of 4 plus 3 space sinx end root end fraction dx$
Put $4 plus 3 space sinx equals space straight y comma space space space space therefore space space 3 space cosx space dx space equals space dy space space space space rightwards double arrow space space space cosxdx space equals space 1 third dy$
When x = 0, y = 4 + 3 sin 0 = 4 + 0 = 4
When x = 2 $straight pi$ , y = 4 + 3 sin 2 $straight pi$ = 4 + 3 x 0 = 4 + 0 = 4
$therefore$ $straight I space equals space 1 third integral subscript 4 superscript 4 fraction numerator 1 over denominator square root of straight y end fraction dy space equals space 1 third integral subscript 4 superscript 4 straight y to the power of negative 1 half end exponent dy space equals space 1 third open square brackets fraction numerator straight y to the power of begin display style 1 half end style end exponent over denominator begin display style 1 half end style end fraction close square brackets subscript 4 superscript 4 space equals space 2 over 3 open square brackets square root of straight y close square brackets subscript 4 superscript 4 space equals space 2 over 3 left parenthesis 2 minus 2 right parenthesis space equals space 2 over 3 cross times 0 space equals 0$

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Integrals

Evaluate $integral subscript 0 superscript 2 straight pi end superscript fraction numerator cosx over denominator square root of 4 plus 3 sinx end root end fraction dx.$

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