Question

Evaluate $space integral subscript 0 superscript 3 square root of straight pi squared end root end superscript square root of straight x space cos squared open parentheses straight x to the power of 3 over 2 end exponent close parentheses dx.$

Solution

Let I = $integral subscript 0 superscript cube root of straight pi squared end root end superscript square root of straight x space cos squared open parentheses straight x to the power of 3 over 2 end exponent close parentheses dx$
Put $straight x to the power of 3 over 2 end exponent space equals space straight y comma space space space space therefore space space space 3 over 2 straight x to the power of 1 half end exponent space equals space dy space space space space rightwards double arrow space space space space space square root of straight x space dx space equals space 2 over 3 dy$
When x = 0, y = 0
When $straight x equals space left parenthesis straight pi squared right parenthesis to the power of 1 third end exponent space straight y space equals space open square brackets open parentheses straight pi squared close parentheses to the power of 1 third end exponent close square brackets to the power of 3 over 2 end exponent space equals space open parentheses straight pi to the power of 2 over 3 end exponent close parentheses to the power of 3 over 2 end exponent space equals space straight pi$
$therefore$ I = $2 over 3 integral subscript 0 superscript straight pi cos squared space straight y space dy space equals space 1 third integral subscript 0 superscript straight pi left parenthesis 2 space cos squared space straight y right parenthesis dy space equals space 1 third integral subscript 0 superscript straight pi left parenthesis 1 plus cos space 2 straight y right parenthesis space dy$
$equals space 1 third open square brackets straight y plus fraction numerator sin space 2 straight y over denominator 2 end fraction close square brackets subscript 0 superscript straight pi space equals space 1 third open square brackets open parentheses straight pi plus fraction numerator sin space 2 straight pi over denominator 2 end fraction close parentheses minus open parentheses 0 plus fraction numerator sin space 0 over denominator 2 end fraction close parentheses close square brackets equals space 1 third open square brackets open parentheses straight pi plus 0 over 2 close parentheses minus open parentheses 0 plus 0 over 2 close parentheses close square brackets space equals space 1 third left square bracket straight pi right square bracket space equals space space straight pi over 3$

For Daily Free Study Material Join wiredfaculty

Integrals

Evaluate $space integral subscript 0 superscript 3 square root of straight pi squared end root end superscript square root of straight x space cos squared open parentheses straight x to the power of 3 over 2 end exponent close parentheses dx.$

Some More Questions From Integrals Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction