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

90 Views

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

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

113 Views

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

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

97 Views

Short Answer Type

86.
Evaluate $integral subscript 0 superscript 2 straight pi end superscript fraction numerator cosx over denominator square root of 4 plus 3 sinx end root end fraction dx.$

Let I = $integral subscript 0 superscript 2 straight pi end superscript fraction numerator cos space straight x over denominator square root of 4 plus 3 space sinx end root end fraction dx$
Put $4 plus 3 space sinx equals space straight y comma space space space space therefore space space 3 space cosx space dx space equals space dy space space space space rightwards double arrow space space space cosxdx space equals space 1 third dy$
When x = 0, y = 4 + 3 sin 0 = 4 + 0 = 4
When x = 2 $straight pi$ , y = 4 + 3 sin 2 $straight pi$ = 4 + 3 x 0 = 4 + 0 = 4
$therefore$ $straight I space equals space 1 third integral subscript 4 superscript 4 fraction numerator 1 over denominator square root of straight y end fraction dy space equals space 1 third integral subscript 4 superscript 4 straight y to the power of negative 1 half end exponent dy space equals space 1 third open square brackets fraction numerator straight y to the power of begin display style 1 half end style end exponent over denominator begin display style 1 half end style end fraction close square brackets subscript 4 superscript 4 space equals space 2 over 3 open square brackets square root of straight y close square brackets subscript 4 superscript 4 space equals space 2 over 3 left parenthesis 2 minus 2 right parenthesis space equals space 2 over 3 cross times 0 space equals 0$

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87.

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

97 Views

88.

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

94 Views

89.

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

152 Views

90.

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