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Integrals

Short Answer Type

241.

If space straight f left parenthesis straight x right parenthesis space equals space integral subscript 0 superscript straight x straight t space sint space dt comma space write space the space value space of space straight f apostrophe left parenthesis straight x right parenthesis

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242.

If $integral subscript 0 superscript straight a fraction numerator 1 over denominator 4 plus straight x squared end fraction dx equals straight pi over 8$ , find the value of a.

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243.

Evaluate colon integral subscript 0 superscript straight pi fraction numerator 4 straight x space sinx over denominator 1 plus cos squared straight x end fraction dx

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244.

Evaluate:
$integral fraction numerator straight x plus 2 over denominator square root of straight x squared plus 5 straight x plus 6 end root end fraction dx$

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Long Answer Type

245.

Evaluate:
$integral fraction numerator 1 over denominator sin to the power of 4 straight x plus sin squared xcos squared straight x plus cos to the power of 4 straight x end fraction dx$

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Short Answer Type

246.
Evaluate: $integral fraction numerator sin left parenthesis straight x minus straight a right parenthesis over denominator sin left parenthesis straight x plus straight a right parenthesis end fraction dx$

straight I space equals space integral fraction numerator sin left parenthesis straight x minus straight a right parenthesis over denominator sin left parenthesis straight x plus straight a right parenthesis 1 end fraction dx

Let space left parenthesis straight x plus straight a right parenthesis space equals space straight t space rightwards double arrow space dx space equals space dt therefore space straight I space equals space integral fraction numerator sin left parenthesis straight t minus 2 straight a right parenthesis over denominator sint end fraction dt equals space integral fraction numerator sint space cos 2 straight a minus cost space sin 2 straight a over denominator sin space straight t end fraction dt equals integral left parenthesis cos 2 straight a minus cot space straight t space sin 2 straight a right parenthesis space dt equals left parenthesis cos 2 straight a right parenthesis straight t minus sin 2 straight a space log open vertical bar sin space straight t close vertical bar plus straight C equals cos 2 straight a left parenthesis straight x plus straight a right parenthesis minus sin 2 straight a space log open vertical bar sin left parenthesis straight x plus straight a right parenthesis close vertical bar plus straight C

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247.

Evaluate:
$integral fraction numerator 5 straight x minus 2 over denominator left parenthesis 1 plus 2 straight x plus 3 straight x squared end fraction dx$

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248.

Evaluate:
$integral fraction numerator straight x squared over denominator left parenthesis straight x squared plus 4 right parenthesis left parenthesis straight x squared plus 9 right parenthesis end fraction dx$

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249.

Evaluate: $integral subscript 0 superscript 4 open parentheses open vertical bar straight x close vertical bar plus open vertical bar straight x minus 2 close vertical bar plus open vertical bar straight x minus 4 close vertical bar close parentheses dx$