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

108 Views

94.
Evaluate the following definite integral:
$integral subscript 1 superscript 2 fraction numerator 1 over denominator straight x left parenthesis 1 plus log space straight x right parenthesis squared end fraction dx$

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

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

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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

94. Evaluate the following definite integral:

94.
Evaluate the following definite integral:
$integral subscript 1 superscript 2 fraction numerator 1 over denominator straight x left parenthesis 1 plus log space straight x right parenthesis squared end fraction dx$