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

Let I = $integral subscript straight pi over 6 end subscript superscript straight pi over 2 end superscript fraction numerator cosecx space cosx over denominator 1 plus cosec squared straight x end fraction dx$
Put cosec x = y, $therefore space space space space minus cosecx space cot space straight x space dx space equals space dy$ $rightwards double arrow space space cosec space straight x space cotx space dx space equals space minus dy$
When $straight x equals straight pi over 6 comma space space space space space straight y space equals space cosec straight pi over 6 space equals space 2$

When $straight x equals straight pi over 2$ $straight y equals cosec straight pi over 2 space equals space 1$

$therefore space space space straight I space equals space space integral subscript 2 superscript 1 fraction numerator 1 over denominator 1 plus straight y squared end fraction dy space equals space minus open square brackets tan to the power of negative 1 end exponent space straight y close square brackets subscript 2 superscript 1 space equals space minus left parenthesis tan to the power of negative 1 end exponent 1 space minus space tan to the power of negative 1 end exponent 2 right parenthesis space equals space tan to the power of negative 1 end exponent 2 space minus space tan to the power of negative 1 end exponent 1$

$equals space tan to the power of negative 1 end exponent open parentheses fraction numerator 2 minus 1 over denominator 1 plus 2 end fraction close parentheses space equals space tan to the power of negative 1 end exponent 1 third$

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

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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

96. Evaluate the following integral:

96.
Evaluate the following integral:
$integral subscript straight pi over 6 end subscript superscript straight pi over 2 end superscript fraction numerator cosec space straight x space cotx over denominator 1 minus cosec squared straight x end fraction dx$