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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.

Let I = $integral subscript 0 superscript 1 fraction numerator straight e to the power of straight x over denominator 1 plus straight e to the power of 2 straight x end exponent end fraction dx$
Put $straight e to the power of straight x space equals space straight y comma space space space space space space space space space space space space therefore space space straight e to the power of straight x dx space equals space dy$
When $straight x equals space 0 comma space space space space space straight y space equals space straight e to the power of 0 space equals space 1$
When $straight x space equals space 1 comma space space space space straight y space equals space straight e to the power of 1 space equals space straight e$
$therefore space space space space space space straight I space equals space integral subscript 1 superscript straight e fraction numerator 1 over denominator 1 plus straight y squared end fraction dy space equals space open square brackets tan to the power of negative 1 end exponent space straight y close square brackets subscript 1 superscript straight e space equals space tan to the power of negative 1 end exponent straight e minus tan to the power of negative 1 end exponent 1 space equals space tan to the power of negative 1 end exponent open parentheses fraction numerator straight e minus 1 over denominator 1 plus straight e end fraction close parentheses$

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93.

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

108 Views

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

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

110 Views

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

92. Evaluate the following integral using substitution.

92.
Evaluate the following integral using substitution.