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

We need to evaluate the integral
$integral fraction numerator straight x plus 2 over denominator square root of straight x squared plus 5 straight x plus 6 end root end fraction dx Let space straight I space equals space integral fraction numerator straight x plus 2 over denominator square root of straight x squared plus 5 straight x plus 6 end root end fraction dx Consider space the space integral space as space follows colon fraction numerator straight x plus 2 over denominator square root of straight x squared plus 5 straight x plus 6 end root end fraction space equals fraction numerator straight A begin display style straight d over dx end style left parenthesis straight x squared plus 5 straight x plus 6 right parenthesis plus straight B over denominator square root of straight x squared plus 5 straight x plus 6 end root end fraction rightwards double arrow straight x plus 2 space equals space straight A left parenthesis 2 straight x plus 5 right parenthesis plus straight B rightwards double arrow straight x plus 2 equals left parenthesis 2 straight A right parenthesis straight x plus 5 straight A plus straight B$
Comparing the coefficients, we have
2A = 1; 5A+B = 2
Solving the above equations, we have

$straight A equals 1 half space and space straight B equals space minus 1 half Thus comma space straight I space equals space integral fraction numerator straight x plus 2 over denominator square root of straight x squared plus 5 straight x plus 6 end root end fraction dx space space equals space integral fraction numerator begin display style fraction numerator 2 straight x plus 5 over denominator 2 end fraction end style minus begin display style 1 half end style over denominator square root of straight x squared plus 5 straight x plus 6 end root end fraction dx space space equals 1 half integral fraction numerator 2 straight x plus 5 over denominator square root of straight x squared plus 5 straight x plus 6 end root end fraction dx minus 1 half integral fraction numerator 1 over denominator square root of straight x squared plus 5 straight x plus 6 end root end fraction dx straight I space equals 1 half straight I subscript 1 space minus space 1 half straight I subscript 2 comma where space straight I subscript 1 space equals integral fraction numerator 2 straight x plus 5 over denominator square root of straight x squared plus 5 straight x plus 6 end root end fraction dx$

$and space straight I subscript 2 equals space integral fraction numerator 1 over denominator square root of straight x squared plus 5 straight x plus 6 end root end fraction dx Now space consider space straight I subscript 1 colon straight I subscript 1 space equals space integral fraction numerator 2 straight x plus 5 over denominator square root of straight x squared plus 5 straight x plus 6 end root end fraction dx Susbstitute straight x squared plus 5 straight x plus 6 space equals straight t semicolon space left parenthesis 2 straight x plus 5 right parenthesis dx space equals space dt straight I subscript 1 space equals integral fraction numerator dt over denominator square root of straight t end fraction space equals 2 square root of straight t equals 2 square root of straight x squared plus 5 straight x plus 6 end root$
Now consider I₂:
$straight I subscript 2 space equals space integral fraction numerator 1 over denominator square root of straight x squared plus 5 straight x plus 6 end root end fraction dx equals space integral fraction numerator 1 over denominator square root of straight x squared plus 5 straight x plus open parentheses begin display style 5 over 2 end style close parentheses squared plus 6 minus open parentheses begin display style 5 over 2 end style close parentheses squared end root end fraction dx equals integral fraction numerator 1 over denominator square root of open parentheses straight x plus begin display style 5 over 2 end style close parentheses squared plus 6 minus begin display style 25 over 4 end style end root end fraction dx equals space integral fraction numerator 1 over denominator square root of open parentheses straight x plus begin display style 5 over 2 end style close parentheses squared minus begin display style 1 fourth end style end root end fraction dx straight I subscript 2 space equals log open vertical bar straight x plus 5 over 2 minus square root of straight x squared plus 5 straight x plus 6 end root close vertical bar plus straight C Thus comma space straight I equals 1 half straight I subscript 1 minus 1 half straight I subscript 2 straight I space equals space square root of straight x squared plus 5 straight x plus 6 end root minus 1 half log open vertical bar straight x plus 5 over 2 minus square root of straight x squared plus 5 straight x plus 6 end root close vertical bar plus straight C$

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

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