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Integrals

Long Answer Type

211.

Evaluate:
$integral subscript 0 superscript straight pi fraction numerator straight x over denominator 1 plus sin squared straight x end fraction dx.$

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Short Answer Type

212. By using the properties of definite integrals, evaluate the following:

integral subscript 0 superscript straight pi log space left parenthesis 1 plus cosx right parenthesis space dx

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213. By using the properties of definite integrals, evaluate the following:

integral subscript 0 superscript straight pi over 2 end superscript fraction numerator straight x space sinx space cosx over denominator sin to the power of 4 straight x plus cos to the power of 4 straight x end fraction dx

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Long Answer Type

214. By using the properties of definite integrals, evaluate the following:

integral subscript 0 superscript straight pi fraction numerator straight x space dx over denominator straight a squared space cos squared straight x plus straight b squared space sin squared straight x end fraction

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Short Answer Type

215.

Show that:
$integral subscript 0 superscript straight a fraction numerator square root of straight x over denominator square root of straight x plus square root of straight a minus straight x end root end fraction space equals space straight a over 2$

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216.

Show that:
$integral subscript 0 superscript straight a fraction numerator 1 over denominator straight x plus square root of straight a squared minus straight x squared end root end fraction dx space equals space straight pi over 4$

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217.
Evaluate $integral subscript 1 superscript 4 straight f left parenthesis straight x right parenthesis space dx space space where space straight f left parenthesis straight x right parenthesis space equals space open vertical bar straight x minus 1 close vertical bar space plus space open vertical bar straight x minus 2 close vertical bar space plus space open vertical bar straight x minus 3 close vertical bar$

Here, $straight f left parenthesis straight x right parenthesis space equals space open vertical bar straight x minus 1 close vertical bar space plus space open vertical bar straight x minus 2 close vertical bar plus space open vertical bar straight x minus 3 close vertical bar$
Let I = $integral subscript 1 superscript 4 straight f left parenthesis straight x right parenthesis space dx space equals space integral subscript 1 superscript 2 straight f left parenthesis straight x right parenthesis space dx plus integral subscript 2 superscript 3 straight f left parenthesis straight x right parenthesis space dx plus integral subscript 3 superscript 4 straight f left parenthesis straight x right parenthesis space dx$
$equals space integral subscript 1 superscript 2 open square brackets left parenthesis straight x minus 1 right parenthesis space minus space left parenthesis straight x minus 2 right parenthesis space minus space left parenthesis straight x minus 3 right parenthesis close square brackets dx space plus space integral subscript 2 superscript 3 open square brackets left parenthesis straight x minus 1 right parenthesis space plus left parenthesis straight x minus 2 right parenthesis space minus space left parenthesis straight x minus 3 right parenthesis close square brackets dx$
$plus integral subscript 3 superscript 4 open square brackets left parenthesis straight x minus 1 right parenthesis plus left parenthesis straight x minus 2 right parenthesis plus left parenthesis straight x minus 3 right parenthesis close square brackets dx$
$equals space integral subscript 1 superscript 2 left parenthesis negative straight x plus 4 right parenthesis space dx plus space integral subscript 2 superscript 3 straight x space dx space plus integral subscript 3 superscript 4 left parenthesis 3 straight x minus 6 right parenthesis space dx equals space open square brackets negative straight x squared over 2 plus 4 straight x close square brackets subscript 1 superscript 2 space plus space open square brackets straight x squared over 2 close square brackets subscript 2 superscript 3 plus open square brackets fraction numerator 3 straight x squared over denominator 2 end fraction minus 6 straight x close square brackets subscript 3 superscript 4 equals left parenthesis negative 2 plus 8 right parenthesis space space minus space open parentheses negative 1 half plus 4 close parentheses plus open parentheses 9 over 2 minus 2 close parentheses space plus space left parenthesis 24 minus 24 right parenthesis space minus space open parentheses 27 over 2 minus 18 close parentheses equals space 6 minus 7 over 2 plus 5 over 2 plus 0 plus 9 over 2 space equals space 19 over 2$

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218.

Evaluate: $integral subscript negative 5 end subscript superscript 0 space straight f left parenthesis straight x right parenthesis space dx comma space space where space straight f left parenthesis straight x right parenthesis space equals space open vertical bar straight x close vertical bar plus open vertical bar straight x plus 2 close vertical bar plus open vertical bar straight x plus 5 close vertical bar.$

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219.

Evaluate: $integral subscript negative 5 end subscript superscript 0 straight f left parenthesis straight x right parenthesis space dx comma space space space where space straight f left parenthesis straight x right parenthesis space equals space open vertical bar straight x close vertical bar plus open vertical bar straight x plus 3 close vertical bar plus open vertical bar straight x plus 6 close vertical bar.$

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220.

If [ x ] stands for integral part of x, then show that $integral subscript 0 superscript 1 left square bracket 5 space straight x right square bracket space dx space equals space 2.$

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Previous Year Papers

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Long Answer Type

Short Answer Type

Long Answer Type

Short Answer Type

217. Evaluate