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

Let I = $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sinx space over denominator 1 plus cos squared straight x end fraction dx$
Put cos x = t, ∴ sin x dx = dt ⇒ sin x dx = – dt When x = 0, t = cos 0 = 1
When x = $straight pi over 2 comma space space straight t space equals space cos straight pi over 2 space equals space 0$
$therefore space space space space space space straight I space equals space minus integral subscript 1 superscript 0 fraction numerator dt over denominator 1 plus straight t squared end fraction space equals space minus left square bracket tan to the power of negative 1 end exponent straight t right square bracket subscript 1 superscript 0 space equals space minus left square bracket tan to the power of negative 1 end exponent 0 minus tan to the power of negative 1 end exponent 1 right square bracket space equals space minus open square brackets 0 minus straight pi over 4 close square brackets space equals space straight pi over 4$

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96.

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

110 Views

97.

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

120 Views

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

104 Views

Previous Year Papers

Test Series

Test Yourself

Short Answer Type

95. Evaluate the following integral using substitution

95.
Evaluate the following integral using substitution
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator sin space straight x over denominator 1 plus cos squared straight x end fraction dx$