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Integrals

Question
CBSEENMA12032413
Wired Faculty App

Evaluate the following:
integral subscript 0 superscript straight pi over 4 end superscript space tan to the power of 4 space 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 tan squared straight x space dx
         equals space integral subscript 0 superscript straight pi over 4 end superscript space tan squared straight x 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 straight x space sec squared straight x minus tan squared straight x right parenthesis space dx
        equals space integral subscript 0 superscript straight pi over 4 end superscript space 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 space 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 0 plus 0 close parentheses equals space open parentheses 1 third cross times 1 minus 1 plus straight pi over 4 close parentheses minus left parenthesis 0 plus 0 plus 0 right parenthesis space equals space 1 third minus 1 plus straight pi over 4 equals straight pi over 4 minus 2 over 3

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