Evaluate the following:
$integral subscript 0 superscript 2 straight pi end superscript straight e to the power of straight x space sin open parentheses straight pi over 4 plus straight x over 2 close parentheses dx.$

Solution

Let
$straight I space equals space integral subscript 0 superscript 2 straight pi end superscript straight e to the power of straight x space sin open parentheses straight pi over 4 plus straight x over 2 close parentheses dx$
Put $straight pi over 4 plus straight x over 2 space equals space straight t comma space space space space space space therefore space space space 1 half dx space equals space dt space space space space space space space rightwards double arrow space space space dx space equals space 2 space dt$
When $straight x space equals space 0 comma space space space straight t space equals space straight pi over 4$
When $straight x space equals space 2 straight pi comma space space space straight t space equals space fraction numerator 5 straight pi over denominator 4 end fraction$
$therefore$ $straight I space equals space integral subscript straight pi over 4 end subscript superscript fraction numerator 5 straight pi over denominator 4 end fraction end superscript space straight e to the power of 2 straight t minus straight pi over 2 end exponent space sin space straight t space dt space equals space 2 space straight e to the power of straight pi over 2 end exponent integral subscript straight pi over 4 end subscript superscript fraction numerator 5 straight pi over denominator 4 end fraction end superscript straight e to the power of 2 straight t end exponent space sint space dt$
$equals space 2 straight e to the power of negative straight pi over 2 end exponent. fraction numerator 1 over denominator 4 plus 1 end fraction open square brackets straight e to the power of 2 straight t end exponent left parenthesis 2 space sint space minus space cost right parenthesis close square brackets subscript straight pi over 4 end subscript superscript fraction numerator 5 straight pi over denominator 4 end fraction end superscript$
$equals space space 2 over 5 straight e to the power of negative straight pi over 2 end exponent open square brackets straight e to the power of fraction numerator 5 straight pi over denominator 2 end fraction end exponent open parentheses 2 space sin space fraction numerator 5 straight pi over denominator 4 end fraction minus cos fraction numerator 5 straight pi over denominator 4 end fraction close parentheses space minus space straight e to the power of straight pi over 2 end exponent open parentheses 2 sin straight pi over 4 minus cos straight pi over 4 close parentheses close square brackets space equals space 2 over 5 straight e to the power of negative straight pi over 2 end exponent space open square brackets straight e to the power of fraction numerator 5 space straight pi over denominator 2 end fraction end exponent open parentheses negative fraction numerator 2 over denominator square root of 2 end fraction plus fraction numerator 1 over denominator square root of 2 end fraction close parentheses space minus straight e to the power of straight pi over 2 end exponent open parentheses fraction numerator 2 over denominator square root of 2 end fraction minus fraction numerator 1 over denominator square root of 2 end fraction close parentheses close square brackets space equals 2 over 5 open square brackets straight e to the power of 2 straight pi end exponent open parentheses negative fraction numerator 1 over denominator square root of 2 end fraction close parentheses minus fraction numerator 1 over denominator square root of 2 end fraction close square brackets space equals space minus fraction numerator square root of 2 over denominator 5 end fraction left parenthesis straight e to the power of 2 straight pi end exponent plus 1 right parenthesis$

For Daily Free Study Material Join wiredfaculty

Integrals

Evaluate the following:
$integral subscript 0 superscript 2 straight pi end superscript straight e to the power of straight x space sin open parentheses straight pi over 4 plus straight x over 2 close parentheses dx.$

Some More Questions From Integrals Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction