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Integrals

Short Answer Type

221.

Show that $integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space straight g left parenthesis straight x right parenthesis space dx space equals space 2 space integral subscript 0 superscript straight a straight f left parenthesis straight x right parenthesis space dx comma space space if space straight f$ and g are defined as f(x) = f(a - x) and g(x) + g(a-x) = 4

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Long Answer Type

222.

Evaluate:
$integral subscript 0 superscript 1 cot to the power of negative 1 end exponent left parenthesis 1 minus straight x plus straight x squared right parenthesis dx.$

142 Views

Multiple Choice Questions

223.
The value of $integral subscript negative straight pi over 2 end subscript superscript straight pi over 2 end superscript left parenthesis straight x cubed plus straight x space cosx space plus space tan to the power of 5 straight x plus 1 right parenthesis space dx space is$

0

2

$straight pi$
$straight pi$

straight pi

Let
$straight I space equals space integral subscript negative straight pi over 2 end subscript superscript straight pi over 2 end superscript space left parenthesis straight x cubed plus straight x space cosx space plus tan to the power of 5 straight x space plus space 1 right parenthesis dx$

$therefore space space space space space space space space equals space integral subscript negative straight pi over 2 end subscript superscript straight pi over 2 end superscript left parenthesis straight x cubed plus straight x space cosx plus tan to the power of 5 straight x right parenthesis space dx space plus space integral subscript straight pi over 2 end subscript superscript straight pi over 2 end superscript space 1 space dx space equals space 0 plus open square brackets straight x close square brackets subscript negative straight pi over 2 end subscript superscript straight pi over 2 end superscript open square brackets because space space straight x cubed plus straight x space cosx plus space tan to the power of 5 straight x space is space odd space function space and space so space integral subscript negative straight pi over 2 end subscript superscript bevelled straight pi over 2 end superscript left parenthesis straight x cubed plus straight x space cosx plus space tan to the power of 5 straight x right parenthesis space dx space equals space 0 close square brackets space space space space space space equals space straight pi over 2 minus open parentheses negative straight pi over 2 close parentheses space equals space straight pi over 2 plus straight pi over 2 space equals space straight pi$
$therefore space space space space left parenthesis straight C right parenthesis space is space correct space answer.$

125 Views

224.

The value of $integral subscript 0 superscript straight pi over 2 end superscript space log space space open parentheses fraction numerator 4 plus 3 space sinx over denominator 4 plus 3 space cosx end fraction close parentheses dx$ is

2
$3 over 4$
0
0

139 Views

225.

If $straight f left parenthesis straight a plus straight b minus straight x right parenthesis space equals space straight f left parenthesis straight x right parenthesis comma space space then space integral subscript straight a superscript straight b straight x space straight f left parenthesis straight x right parenthesis space dx$ is equal to

$fraction numerator straight a plus straight b over denominator 2 end fraction integral subscript straight a superscript straight b straight f left parenthesis straight b minus straight x right parenthesis space dx$
$fraction numerator straight a plus straight b over denominator 2 end fraction integral subscript straight a superscript straight b straight f left parenthesis straight b plus straight x right parenthesis space dx$
$fraction numerator straight b minus straight a over denominator 2 end fraction integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx$
$fraction numerator straight b minus straight a over denominator 2 end fraction integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx$

108 Views

226.

The value of $integral subscript 0 superscript 1 space tan to the power of negative 1 end exponent open parentheses fraction numerator 2 straight x minus 1 over denominator 1 plus straight x minus straight x squared end fraction close parentheses dx$ is

1
0
-1
-1

137 Views

Long Answer Type

227.

Evaluate $integral subscript 0 superscript straight pi over 2 end superscript log space sinx space dx.$

111 Views

Short Answer Type

228.

Evaluate the definite integral:
$integral subscript 2 superscript 3 fraction numerator 1 over denominator straight x squared minus 1 end fraction dx$

124 Views

229.

Evaluate the definite integral:
$integral subscript 0 superscript 1 divided by square root of 2 end superscript fraction numerator sin to the power of negative 1 end exponent straight x over denominator left parenthesis 1 minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction dx$

181 Views

Long Answer Type

230.

Evaluate:
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator straight a space cosx plus straight b space sinx end fraction. space straight a comma space straight b space greater than space 0$

195 Views

Previous Year Papers

Test Series

Test Yourself

Short Answer Type

Long Answer Type

Multiple Choice Questions

223. The value of 0 2

Long Answer Type

Short Answer Type

Long Answer Type