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Integrals

Short Answer Type

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

Since differentiation operation is the inverse operation of integration, we have
$straight f apostrophe left parenthesis straight x right parenthesis space equals space straight x space sinx Let space straight f left parenthesis straight x right parenthesis space equals space integral subscript 0 superscript straight x straight t space sint space dt$
Let us do this by integration by parts.
Therefore assume u = t; du = dt

$integral sint space dt space space equals space integral dv minus cost space equals straight v Therefore comma space straight f left parenthesis straight x right parenthesis space equals space open square brackets straight t left parenthesis negative cost right parenthesis close square brackets subscript 0 superscript straight x space minus space integral subscript 0 superscript straight x left parenthesis negative cos space straight t right parenthesis space dt straight f left parenthesis straight x right parenthesis space equals space minus straight x space cosx space plus space sinx space plus straight C$
Differentiating the above function with respect to x,

$straight f apostrophe left parenthesis straight x right parenthesis space equals space minus open square brackets straight x left parenthesis negative sinx right parenthesis minus sinx close square brackets space plus space sinx space equals space straight x space sinx$

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

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

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250.

Evaluate: $integral subscript 2 superscript 3 space 3 to the power of straight x space dx$ .

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