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Integrals

Short Answer Type

181.

By using the properties of definite integrals, evaluate the following integral:
$integral subscript 0 superscript 4 open vertical bar straight x minus 1 close vertical bar space dx$

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182.

By using the properties of definite integrals, evaluate the following integral:
$integral subscript 2 superscript 5 open vertical bar straight x minus 1 close vertical bar dx$

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183.

Show that:
$integral subscript 1 superscript 5 open vertical bar straight x minus 3 close vertical bar space dx space equals space 4$

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184.

Show that:
$integral subscript 0 superscript 1 open vertical bar 3 straight x minus 1 close vertical bar space dx space equals space 5 over 6$

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185.

By using the properties of definite integrals, evaluate the following integral:

$integral subscript negative 5 end subscript superscript 5 open vertical bar straight x plus 2 close vertical bar dx$

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186.

By using the properties of definite integrals, evaluate the following integral:
$integral subscript 2 superscript 8 open vertical bar straight x minus 5 close vertical bar space dx$

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Long Answer Type

187.

Show that:
$integral subscript negative 1 end subscript superscript 2 open vertical bar straight x cubed minus straight x close vertical bar space dx.$

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Short Answer Type

188.

Evaluate: $integral subscript 0 superscript 1 open vertical bar 2 straight x minus 1 close vertical bar space dx.$

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189.
Show that:
$integral subscript 0 superscript straight pi open vertical bar cos space straight x close vertical bar space dx space equals 2$

Let I = $integral subscript 0 superscript straight pi open vertical bar cos space straight x close vertical bar space dx$
For $0 less or equal than straight x less or equal than straight pi over 2 space cosx space greater or equal than 0 space rightwards double arrow space space open vertical bar cos space straight x close vertical bar space equals space cos space straight x$
and for $straight pi over 2 less or equal than straight x less or equal than straight pi comma space space space cos space straight x space less or equal than 0 space space rightwards double arrow space space space open vertical bar cos space straight x close vertical bar space equals space minus cos space straight x$
$therefore space space space straight I space equals integral subscript 0 superscript straight pi over 2 end superscript open vertical bar cos space straight x close vertical bar dx space plus space integral subscript straight pi over 2 end subscript superscript straight pi open vertical bar cosx close vertical bar dx space equals space integral subscript 0 superscript straight pi over 2 end superscript cosx space dx space minus space integral subscript straight pi over 2 end subscript superscript straight pi cosx space dx$
$equals space open square brackets sin space straight x close square brackets subscript 0 superscript straight pi over 2 end superscript space minus space open square brackets sin space straight x close square brackets subscript straight pi over 2 end subscript superscript straight pi space equals space open parentheses sin straight pi over 2 minus sin space 0 close parentheses space minus space open square brackets sin space straight pi space minus space sin space straight pi over 2 close square brackets equals space left parenthesis 1 minus 0 right parenthesis space minus space left parenthesis 0 minus 1 right parenthesis space equals space 2$

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190.

Show that:
$integral subscript 0 superscript straight pi over 2 end superscript space open vertical bar sin space straight x space cosx close vertical bar space dx space equals space 1 half$