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

113 Views

83.

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

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

Let I = $integral subscript 0 superscript straight pi over 6 end superscript space cos to the power of negative 3 end exponent 2 straight theta space sin 2 straight theta space space dθ space equals space integral subscript 0 superscript straight pi over 6 end superscript fraction numerator sin space 2 straight theta space dθ over denominator cos cubed space 2 straight theta end fraction$
Put $cos space 2 straight theta space equals space straight y comma space space therefore space space minus 2 space sin space 2 straight theta space dθ space equals space dy space space space rightwards double arrow space space sin space 2 straight theta space dθ space equals space minus 1 half dy$
When $straight theta space equals space 0 comma space space straight y space equals space cos space 0 space equals space 1$
When $straight theta space equals space straight pi over 6 comma space space space straight y space equals space cos straight pi over 3 space equals space 1 half$
$therefore$ $straight I space equals space minus 1 half integral subscript 1 superscript 1 half end superscript 1 over straight y cubed dy space equals space minus 1 half integral subscript 1 superscript 1 half end superscript straight y to the power of negative 3 end exponent space dy space equals space minus 1 half open square brackets fraction numerator straight y to the power of negative 2 end exponent over denominator negative 2 end fraction close square brackets to the power of 1 half end exponent$

$equals space 1 fourth open square brackets 1 over straight y squared close square brackets subscript 1 superscript 1 half end superscript space equals space 1 fourth left square bracket 4 minus 1 right square bracket space equals space 1 fourth cross times 3 space equals space 3 over 4$

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90.

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