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Integrals

Short Answer Type

91.

Evaluate the following integral using substitution.
$integral subscript 0 superscript 1 fraction numerator straight x over denominator straight x squared plus 1 end fraction dx$

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92.

Evaluate the following integral using substitution.

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93.

Evaluate the following definite integral:
$integral subscript 0 superscript 1 straight x space straight e to the power of straight x squared end exponent space dx$

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94.

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

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95.

Evaluate the following integral using substitution
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x over denominator 1 plus cos squared straight x end fraction dx$

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96.

Evaluate the following integral:
$integral subscript straight pi over 6 end subscript superscript straight pi over 2 end superscript fraction numerator cosec space straight x space cotx over denominator 1 minus cosec squared straight x end fraction dx$

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97.

Evaluate the following integral:
$integral subscript 1 superscript 2 1 over straight x squared straight e to the power of negative 1 over straight x end exponent dx$

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98.
Evaluate the following integral:
$integral subscript 0 superscript 1 fraction numerator left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis squared over denominator 1 plus straight x squared end fraction dx$

Let I = $integral subscript 0 superscript 1 fraction numerator left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis squared over denominator 1 plus straight x squared end fraction dx$
Put $tan to the power of negative 1 end exponent straight x space equals space straight y comma space space space space space therefore space space space space fraction numerator 1 over denominator 1 plus straight x squared end fraction dx space equals space dy$
When x = 0, $straight y space equals tan to the power of negative 1 end exponent 0 space equals space 0$
When x = 1, $straight y space equals space tan to the power of negative 1 end exponent space 1 space space equals space straight pi over 4$
$therefore$ $straight I space equals space integral subscript 0 superscript straight pi over 4 end superscript space straight y squared dy space equals space open square brackets straight y cubed over 3 close square brackets subscript 0 superscript straight pi over 4 end superscript space equals space 1 third open square brackets straight pi cubed over 64 minus 0 close square brackets space equals space straight pi cubed over 192$

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99.

Evaluate the following integral using substitution.
$integral subscript 0 superscript 2 straight x space square root of straight x plus 2 end root dx.$

115 Views

100.

Evaluate $integral subscript negative 1 end subscript superscript 1 5 straight x to the power of 4 square root of straight x to the power of 5 plus 1 end root dx.$

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Previous Year Papers

Test Series

Test Yourself

Short Answer Type

98. Evaluate the following integral:

98.
Evaluate the following integral:
$integral subscript 0 superscript 1 fraction numerator left parenthesis tan to the power of negative 1 end exponent straight x right parenthesis squared over denominator 1 plus straight x squared end fraction dx$