Evaluate $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x space plus space cos space straight x over denominator square root of sinx space cosx end root end fraction dx.$

Let I = $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sinx plus cosx over denominator square root of sin space straight x space cosx end root end fraction dx space equals space 4 integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sinx plus cosx over denominator square root of 2 space sinx space cosx end root end fraction dx$

$equals space 4 space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x space plus space cos space straight x over denominator square root of sin space 2 straight x end root end fraction dx space equals space 4 space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sinx space plus space cosx over denominator square root of 1 minus left parenthesis 1 minus sin space 2 straight x right parenthesis end root end fraction dx$
$equals space 4 space integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x space plus space cos space straight x over denominator square root of 1 minus left parenthesis sin squared straight x plus cos squared straight x minus 2 sinx space cosx right parenthesis end root end fraction dx equals space 4 integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x space plus space cos space straight x space over denominator square root of 1 minus left parenthesis sinx space space minus cosx right parenthesis squared end root end fraction dx$
Put sin x - cos x = y, ∴ (cos x + sin x) dx = dy When x = 0, y = sin 0 – cos 0 = 0 – 1 = – 1
When $straight x equals space straight pi over 2 comma space space space straight y space equals space sin straight pi over 2 minus cos straight pi over 2 space equals space 1 minus 0 space equals space 1$
$therefore$ $straight I space equals space 4 integral subscript negative 1 end subscript superscript 1 fraction numerator 1 over denominator square root of 1 minus straight y squared end root end fraction dy space equals space 4 open square brackets sin to the power of negative 1 end exponent space straight y close square brackets subscript negative 1 end subscript superscript 1 space equals space 4 open square brackets sin to the power of negative 1 end exponent left parenthesis 1 right parenthesis space minus space sin to the power of negative 1 end exponent left parenthesis negative 1 right parenthesis close square brackets$
$equals space 4 space open square brackets straight pi over 2 minus open parentheses negative straight pi over 2 close parentheses close square brackets space equals space 4 open parentheses straight pi over 2 plus straight pi over 2 close parentheses space equals space 4 straight pi$

143 Views

Integrals

Hope you found this question and answer to be good. Find many more questions on Integrals with answers for your assignments and practice.

Mathematics Part II

Browse through more topics from Mathematics Part II for questions and snapshot.