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

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247.
Evaluate:
$integral fraction numerator 5 straight x minus 2 over denominator left parenthesis 1 plus 2 straight x plus 3 straight x squared end fraction dx$

integral fraction numerator 5 straight x minus 2 over denominator 1 plus 2 straight x plus 3 straight x squared end fraction dx equals 5 integral fraction numerator straight x minus begin display style 2 over 5 end style over denominator 1 plus 2 straight x plus 3 straight x squared end fraction dx equals 5 over 6 integral fraction numerator 6 straight x minus begin display style 12 over 5 end style over denominator 1 plus 2 straight x plus 3 straight x squared end fraction dx equals 5 over 6 integral fraction numerator 6 straight x plus 2 minus begin display style 12 over 5 end style minus 2 over denominator 1 plus 2 straight x plus 3 straight x squared end fraction dx equals 5 over 6 integral fraction numerator 6 straight x plus 2 minus begin display style 22 over 5 end style over denominator 1 plus 2 straight x plus 3 straight x squared end fraction dx equals 5 over 6 integral fraction numerator 6 straight x plus 2 over denominator 1 plus 2 straight x plus 3 straight x squared end fraction dx minus 5 over 6 cross times 22 over 5 integral fraction numerator 1 over denominator 3 open curly brackets open parentheses straight x plus begin display style 1 third end style close parentheses squared plus begin display style 2 over 9 end style close curly brackets end fraction dx

equals 5 over 6 log open vertical bar 1 plus 2 straight x plus 3 straight x squared close vertical bar minus 11 over 9 integral fraction numerator 1 over denominator open parentheses straight x plus begin display style 1 third end style close parentheses squared plus begin display style 2 over 9 end style end fraction dx equals 5 over 6 log open vertical bar 1 plus 2 straight x plus 3 straight x squared close vertical bar minus 11 over 9 cross times fraction numerator 3 over denominator square root of 2 end fraction tan to the power of negative 1 end exponent fraction numerator open parentheses straight x plus begin display style 1 third end style close parentheses over denominator begin display style fraction numerator square root of 2 over denominator 3 end fraction end style end fraction plus straight C equals 5 over 6 log open vertical bar 1 plus 2 straight x plus 3 straight x squared close vertical bar minus fraction numerator 11 over denominator 3 square root of 2 end fraction cross times tan to the power of negative 1 end exponent open parentheses fraction numerator 3 straight x plus 1 over denominator square root of 2 end fraction close parentheses plus straight C

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