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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
$straight pi$
$straight pi$

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

Let I = $integral subscript 0 superscript 1 divided by square root of 2 end superscript fraction numerator sin to the power of negative 1 end exponent straight x over denominator left parenthesis 1 minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction dx$
Put $straight x space equals space sin space straight theta space space space or space space straight theta space equals space sin to the power of negative 1 end exponent straight x comma space space space space therefore space space space dx space equals space space cosθ space dθ$
When x = 0, $sin space straight theta space equals space 0 space space space space space space space space space space rightwards double arrow space space space space space space straight theta space space equals 0$
When $straight x space equals space 1 half comma space space sin space straight theta space equals space fraction numerator 1 over denominator square root of 2 end fraction space space space space space space space rightwards double arrow space space space space straight theta space equals space straight pi over 4$
$therefore space space space space straight I space equals space integral subscript 0 superscript straight pi over 4 end superscript fraction numerator straight theta over denominator left parenthesis 1 minus sin squared straight theta right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction cos space straight theta space dθ space equals space integral subscript 0 superscript straight pi over 4 end superscript fraction numerator straight theta space cosθ over denominator cos cubed straight theta space end fraction dθ space equals space integral subscript 0 superscript straight pi over 4 end superscript fraction numerator straight theta over denominator cos squared straight theta end fraction dθ$
$equals space integral subscript 0 superscript straight pi over 4 end superscript space straight theta space sec squared space straight theta space dθ space equals space open square brackets straight theta space tan space straight theta close square brackets subscript 0 superscript straight pi over 4 end superscript space minus space integral subscript 0 superscript straight pi over 4 end superscript 1. space tan space straight theta space dθ equals space open square brackets straight theta space tan space straight theta close square brackets subscript 0 superscript straight pi over 4 end superscript plus open square brackets log space open vertical bar cos space straight theta close vertical bar close square brackets subscript 0 superscript straight pi over 4 end superscript equals space open parentheses straight pi over 4 tan straight pi over 4 minus 0 close parentheses space plus space open square brackets log space open vertical bar cos space straight pi over 4 close vertical bar minus log space left parenthesis cos space 0 right parenthesis close square brackets equals space straight pi over 4 cross times 1 space minus space 0 space plus space log space fraction numerator 1 over denominator square root of 2 end fraction minus space log space 1 space equals space straight pi over 4 plus log space 1 space minus space log space 2 to the power of 1 half end exponent minus 0 space equals space straight pi over 4 minus 1 half log space 2$

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

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Test Yourself

Short Answer Type

Long Answer Type

Multiple Choice Questions

Long Answer Type

Short Answer Type

229. Evaluate the definite integral:

Long Answer Type