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Integrals

Question
CBSEENMA12032411
Wired Faculty App

Evaluate the following
integral subscript 0 superscript straight pi over 4 end superscript space tan to the power of 4 straight x space dx

Solution

Let I = integral subscript 0 superscript straight pi over 4 end superscript space tan to the power of 4 straight x space dx space equals space integral subscript 0 superscript straight pi over 4 end superscript space tan squared straight x. space space tan squared straight x space dx
        equals space integral subscript 0 superscript straight pi over 4 end superscript space tan squared straight x space left parenthesis sec squared straight x minus 1 right parenthesis space dx space equals space integral subscript 0 superscript straight pi over 4 end superscript left parenthesis tan squared xsec squared straight x minus tan squared straight x right parenthesis space dx
         equals space integral subscript 0 superscript straight pi over 4 end superscript left parenthesis tan squared straight x space sec squared straight x minus sec squared straight x plus 1 right parenthesis space dx space equals space open square brackets fraction numerator tan cubed straight x over denominator 3 end fraction minus tanx plus straight x close square brackets subscript 0 superscript straight pi over 4 end superscript
           equals open parentheses 1 third tan cubed straight pi over 4 minus tan straight pi over 4 plus straight pi over 4 close parentheses space minus space open parentheses 1 third tan cubed 0 minus tan space 0 space plus 0 close parentheses equals space open parentheses 1 third cross times 1 minus 1 plus straight pi over 4 close parentheses space minus space left parenthesis 0 plus 0 plus 0 right parenthesis space equals space 1 third minus 1 plus straight pi over 4 space equals space straight pi over 4 minus 2 over 3

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