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Prove the following:
$integral subscript 1 superscript 3 fraction numerator dx over denominator straight x squared left parenthesis straight x plus 1 right parenthesis end fraction space equals space 2 over 3 plus log 2 over 3$

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Short Answer Type

82.

Prove the following:
$integral subscript 0 superscript straight pi over 4 end superscript space 2 space tan cubed straight x space dx space equals space 1 minus log space 2.$

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83.
Prove the following:
$integral subscript 0 superscript 1 square root of fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction dx end root space equals space straight pi over 2 minus 1.$

Let I = $integral subscript 0 superscript 1 square root of fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction end root dx space equals space integral subscript 0 superscript 1 square root of fraction numerator 1 minus straight x over denominator 1 plus straight x end fraction cross times fraction numerator 1 minus straight x over denominator 1 minus straight x end fraction end root dx$
$equals space integral subscript 0 superscript 1 fraction numerator 1 minus straight x over denominator square root of 1 minus straight x squared end root end fraction dx space equals space integral subscript 0 superscript 1 fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction dx space minus space integral subscript 0 superscript 1 fraction numerator straight x over denominator square root of 1 minus straight x squared end root end fraction dx equals space integral subscript 0 superscript 1 fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction dx plus 1 half integral subscript 0 superscript 1 left parenthesis 1 minus straight x squared right parenthesis space to the power of 1 half end exponent left parenthesis negative 2 straight x right parenthesis space dx$
$equals space open square brackets sin to the power of negative 1 end exponent straight x close square brackets subscript 0 superscript 1 space plus space 1 half open square brackets fraction numerator left parenthesis 1 minus straight x squared right parenthesis to the power of begin display style 1 half end style end exponent over denominator begin display style 1 half end style end fraction close square brackets subscript 0 superscript 1 space equals space open square brackets sin to the power of negative 1 end exponent straight x close square brackets subscript 0 superscript 1 plus open square brackets square root of 1 minus straight x squared end root close square brackets subscript 0 superscript 1$

= $left parenthesis sin to the power of negative 1 end exponent 1 minus sin to the power of negative 1 end exponent 0 right parenthesis space plus space left parenthesis square root of 1 minus 1 end root space minus space square root of 1 minus 0 right parenthesis end root space equals space straight pi over 2 minus 0 plus 0 minus 1 space equals space straight pi over 2 minus 1.$

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Multiple Choice Questions

84.

integral subscript 1 superscript square root of 3 end superscript fraction numerator dx over denominator 1 plus straight x squared end fraction

equals

$straight pi over 3$
$fraction numerator 2 straight pi over denominator 3 end fraction$
$straight pi over 6$
$straight pi over 6$

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85.

integral subscript 0 superscript 2 over 3 end superscript fraction numerator dx over denominator 4 plus 9 straight x squared end fraction space equals

$straight pi over 6$
$straight pi over 12$
$straight pi over 24$
$straight pi over 24$

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Short Answer Type

86.

Evaluate $integral subscript 0 superscript 2 straight pi end superscript fraction numerator cosx over denominator square root of 4 plus 3 sinx end root end fraction dx.$

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87.

Evaluate $integral subscript 0 superscript straight pi over 6 end superscript left parenthesis 1 minus cos space 3 straight theta right parenthesis space sin space 3 straight theta space dθ$

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88.

Evaluate $integral subscript 0 superscript straight pi space 5 space left parenthesis 5 minus 4 space cosθ right parenthesis to the power of 1 fourth end exponent space sin space straight theta space dθ$

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89.

Evaluate $integral subscript 0 superscript straight pi over 6 end superscript space cos to the power of negative 3 end exponent 2 straight theta space sin space 2 straight theta space dθ$

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90.

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