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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

146 Views

Long Answer Type

222.
Evaluate:
$integral subscript 0 superscript 1 cot to the power of negative 1 end exponent left parenthesis 1 minus straight x plus straight x squared right parenthesis dx.$

Let
I = $integral subscript 0 superscript 1 cot to the power of negative 1 end exponent left parenthesis 1 minus straight x plus straight x squared right parenthesis dx$
$equals space integral subscript 0 superscript 1 tan to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator 1 minus straight x plus straight x squared end fraction close parentheses dx space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space cot to the power of negative 1 end exponent straight x space equals space tan to the power of negative 1 end exponent 1 over straight x close square brackets$
$equals space integral subscript 0 superscript 1 tan to the power of negative 1 end exponent open square brackets fraction numerator straight x plus left parenthesis 1 minus straight x right parenthesis over denominator 1 minus straight x left parenthesis 1 minus straight x right parenthesis end fraction close square brackets dx space equals space integral subscript 0 superscript 1 open square brackets tan to the power of negative 1 end exponent straight x plus tan to the power of negative 1 end exponent left parenthesis 1 minus straight x right parenthesis close square brackets dx space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space tan to the power of negative 1 end exponent straight x plus tan to the power of negative 1 end exponent straight y space equals space tan to the power of negative 1 end exponent open parentheses fraction numerator straight x plus straight y over denominator 1 minus xy end fraction close parentheses close square brackets equals space integral subscript 0 superscript 1 space tan to the power of negative 1 end exponent xdx space plus space integral subscript 0 superscript 1 tan to the power of negative 1 end exponent left parenthesis 1 minus straight x right parenthesis space dx equals space integral subscript 0 superscript 1 tan to the power of negative 1 end exponent straight x space dx plus integral subscript 0 superscript 1 tan to the power of negative 1 end exponent left curly bracket 1 minus left parenthesis 1 minus straight x right parenthesis right curly bracket space dx equals space integral subscript 0 superscript 1 tan to the power of negative 1 end exponent straight x space dx plus integral subscript 0 superscript 1 tan to the power of negative 1 end exponent straight x space dx space equals space 2 integral subscript 0 superscript 1 tan to the power of negative 1 end exponent straight x. space 1 space dx$
$equals space 2 open square brackets left parenthesis tan to the power of negative 1 end exponent space straight x right parenthesis space left parenthesis straight x right parenthesis close square brackets subscript 0 superscript 1 space minus space 2 integral subscript 0 superscript 1 fraction numerator 1 over denominator 1 plus straight x squared end fraction cross times straight x space dx equals 2 space open square brackets straight x space tan to the power of negative 1 end exponent straight x close square brackets subscript 0 superscript 1 space minus space integral subscript 0 superscript 1 fraction numerator 2 straight x over denominator 1 plus straight x squared end fraction dx space equals space 2 open square brackets straight x space tan to the power of negative 1 end exponent straight x close square brackets subscript 0 superscript 1 space minus space open square brackets log left parenthesis 1 plus straight x right parenthesis squared close square brackets subscript 0 superscript 1 equals space 2 left square bracket tan to the power of negative 1 end exponent space 1 space minus space 0 right square bracket space minus space left square bracket log space 2 space minus space log space 1 right square bracket equals space 2 open parentheses straight pi over 4 minus 0 close parentheses space space minus space left parenthesis log space 2 space minus space 0 right parenthesis space equals space straight pi over 2 minus log space 2$

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

181 Views

Long Answer Type

230.

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