Question

$integral subscript negative 3 straight pi divided by 2 end subscript superscript negative straight pi divided by 2 end superscript open square brackets open parentheses straight x plus straight pi close parentheses cubed plus space cos space left parenthesis straight x space plus 3 straight pi right parenthesis close square brackets dx space is space equal space to$
$straight pi to the power of 4 over 32$
$straight pi to the power of 4 over 32 space plus space pi over 2$
$pi over 2$
$pi over 4 minus 1$

Solution

pi over 2

straight I space equals space integral subscript negative 3 straight pi divided by 2 end subscript superscript straight pi divided by 2 end superscript open square brackets open parentheses straight x plus straight pi close parentheses cubed space plus space cos squared space left parenthesis straight x space plus 3 straight pi right parenthesis close square brackets dx space Put space straight x space plus straight pi space equals space straight t straight I space equals space integral subscript negative straight pi divided by 2 end subscript superscript straight pi divided by 2 end superscript space open square brackets straight t cubed space plus space cos squared straight t close square brackets dt space equals space 2 integral subscript 0 superscript straight pi divided by 2 end superscript space cos squared straight t space dt space equals space integral subscript 0 superscript straight pi divided by 2 end superscript space left parenthesis 1 plus space cos space 2 straight t right parenthesis dt space equals space straight pi over 2 plus 0

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