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

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

Let I = $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator dx over denominator acosx plus bsinx end fraction$

Put $tan space straight x over 2 space equals space straight t space space space or space straight x over 2 space equals space tan to the power of negative 1 end exponent space straight t space rightwards double arrow space space straight x space equals space 2 space tan to the power of negative 1 end exponent straight t space space space space rightwards double arrow space space space dx space equals space fraction numerator 2 over denominator 1 plus straight t squared end fraction dt$
When $straight x space equals space 0 comma space space straight t space equals space tan space 0 space equals space 0$

When $straight x space equals space straight pi over 2 comma space straight t space equals space tan straight pi over 4 space equals space 1$
Also $sinx space equals space fraction numerator 2 space tan space begin display style straight x over 2 end style over denominator 1 plus tan squared begin display style straight x over 2 end style end fraction space equals fraction numerator 2 straight t over denominator 1 minus straight t squared end fraction comma space space cosx space equals space fraction numerator 1 minus tan squared begin display style straight x over 2 end style over denominator 1 plus tan squared begin display style straight x over 2 end style end fraction space equals space fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction$

$therefore space space space space straight I space equals space integral subscript 0 superscript 1 fraction numerator begin display style fraction numerator 2 over denominator 1 plus straight t squared end fraction end style dt over denominator straight a open parentheses begin display style fraction numerator 1 minus straight t squared over denominator 1 plus straight t squared end fraction end style close parentheses plus straight b open parentheses begin display style fraction numerator 2 straight t over denominator 1 plus straight t squared end fraction end style close parentheses end fraction space equals space integral subscript 0 superscript 1 fraction numerator 2 dt over denominator straight a left parenthesis 1 minus straight t squared right parenthesis plus 2 bt end fraction$

$equals space 2 over straight a integral subscript 0 superscript 1 fraction numerator dt over denominator 1 minus straight t squared plus begin display style fraction numerator 2 straight b over denominator straight a end fraction end style straight t end fraction straight t space equals space 2 over straight a integral subscript 0 superscript 1 fraction numerator dt over denominator 1 minus open parentheses straight t squared minus begin display style fraction numerator 2 straight b over denominator straight a end fraction end style straight t close parentheses end fraction equals space 2 over straight a integral subscript 0 superscript 1 fraction numerator dt over denominator open parentheses 1 plus begin display style straight b squared over straight a squared end style close parentheses minus open parentheses straight t squared minus begin display style fraction numerator 2 straight b over denominator straight a end fraction end style straight t plus begin display style straight b squared over straight a squared end style close parentheses end fraction space equals space 2 over straight a integral subscript 0 superscript 1 fraction numerator dt over denominator open parentheses begin display style fraction numerator square root of straight a squared plus straight b squared end root over denominator straight a end fraction end style close parentheses squared minus open parentheses straight t minus begin display style straight b over straight a end style close parentheses squared end fraction equals space 2 over straight a fraction numerator 1 over denominator begin display style fraction numerator 2 square root of straight a squared plus straight b squared end root over denominator straight a end fraction end style end fraction open square brackets fraction numerator open vertical bar log space begin display style fraction numerator square root of straight a squared plus straight b squared end root over denominator straight a end fraction end style plus open parentheses straight t minus begin display style straight b over straight a end style close parentheses close vertical bar over denominator begin display style fraction numerator square root of straight a squared plus straight b squared end root over denominator straight a end fraction end style minus open parentheses straight t minus begin display style straight b over straight a end style close parentheses end fraction close square brackets subscript 0 superscript 1$

$equals space fraction numerator 1 over denominator square root of straight a squared plus straight b squared end root end fraction open square brackets log open square brackets fraction numerator open parentheses begin display style fraction numerator square root of straight a squared plus straight b squared end root over denominator straight a end fraction end style plus 1 minus begin display style straight b over straight a end style close parentheses over denominator begin display style fraction numerator square root of straight a squared plus straight b squared end root over denominator straight a end fraction end style minus 1 plus begin display style straight b over straight a end style end fraction close square brackets minus space log space open parentheses fraction numerator begin display style fraction numerator square root of straight a squared plus straight b squared end root over denominator straight a end fraction minus straight b over straight a end style over denominator begin display style fraction numerator square root of straight a squared plus straight b squared end root over denominator straight a end fraction plus straight b over straight a end style end fraction close parentheses close square brackets space equals space fraction numerator 1 over denominator square root of straight a squared plus straight b squared end root end fraction open square brackets log space open parentheses fraction numerator square root of straight a squared plus straight b squared end root plus straight a minus straight b over denominator square root of straight a squared plus straight b squared end root minus straight a plus straight b end fraction close parentheses minus log space open parentheses fraction numerator square root of straight a squared plus straight b squared end root minus straight b over denominator square root of straight a squared plus straight b squared end root plus straight b end fraction close parentheses close square brackets$

$space equals space fraction numerator 1 over denominator square root of straight a squared plus straight b squared end root end fraction space log space open parentheses fraction numerator square root of straight a squared plus straight b squared end root plus straight a minus straight b over denominator square root of straight a squared plus straight b squared end root minus straight a plus straight b end fraction cross times fraction numerator square root of straight a squared plus straight b squared end root plus straight b over denominator square root of straight a squared plus straight b squared end root minus straight b end fraction close parentheses equals space fraction numerator 1 over denominator square root of straight a squared plus straight b squared end root end fraction log open parentheses fraction numerator straight a squared plus straight b squared plus ab minus straight b squared plus straight a square root of straight a squared plus straight b squared end root over denominator straight a squared plus straight b squared plus ab minus straight b squared minus straight a square root of straight a squared plus straight b squared end root end fraction close parentheses equals space fraction numerator 1 over denominator square root of straight a squared plus straight b squared end root end fraction log open parentheses fraction numerator straight a plus straight b plus square root of straight a squared plus straight b squared end root over denominator straight a plus straight b minus square root of straight a squared plus straight b squared end root end fraction close parentheses$