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

We need to evaluate the integral

$straight I space equals space integral subscript 0 superscript straight pi fraction numerator 4 straight x space sinx over denominator 1 plus cos squared straight x end fraction dx... left parenthesis 1 right parenthesis Using space the space property space integral straight f left parenthesis straight a minus straight x right parenthesis dx equals integral straight f left parenthesis straight x right parenthesis dx comma space we space have straight I equals space integral subscript 0 superscript straight pi fraction numerator 4 left parenthesis straight pi minus straight x right parenthesis space sin left parenthesis straight pi minus straight x right parenthesis over denominator 1 plus cos squared left parenthesis straight pi minus straight x right parenthesis end fraction dx rightwards double arrow space space straight I space equals space integral subscript 0 superscript straight pi fraction numerator 4 straight pi space sinx over denominator 1 plus cos squared straight x end fraction dx minus integral subscript 0 superscript straight pi fraction numerator 4 straight x space sinx over denominator 1 plus cos squared straight x end fraction dx... left parenthesis 2 right parenthesis Adding space equations space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis comma space we space have comma$

$rightwards double arrow 2 straight I space equals space integral subscript 0 superscript straight pi fraction numerator 4 straight x space sinx over denominator 1 plus cos squared straight x end fraction dx space plus space integral subscript 0 superscript straight pi fraction numerator 4 πsinx over denominator 1 plus cos squared straight x end fraction dx minus integral subscript 0 superscript straight pi fraction numerator 4 straight x space sinx over denominator 1 plus cos squared straight x end fraction dx rightwards double arrow 2 straight I space equals space integral subscript 0 superscript straight pi fraction numerator 4 straight pi space sinx over denominator 1 plus cos squared straight x end fraction dx rightwards double arrow 2 straight I space equals space 4 straight pi integral subscript 0 superscript straight pi fraction numerator sinx over denominator 1 plus cos squared straight x end fraction dx$

$Substitute space straight t space equals space cosx semicolon space dt space equals space minus sinxdx when space straight x space equals space 0 comma space space straight t space equals 1 when space straight x space equals space straight pi comma space straight t equals negative 1 rightwards double arrow 2 straight I space equals space 4 straight pi integral subscript 1 superscript negative 1 end superscript fraction numerator left parenthesis negative 1 right parenthesis dt over denominator 1 plus straight t squared end fraction rightwards double arrow space straight I space equals space 2 straight pi integral subscript negative 1 end subscript superscript 1 space fraction numerator dt over denominator 1 plus straight t squared end fraction rightwards double arrow straight I space equals 2 cross times 2 straight pi integral subscript 0 superscript 1 fraction numerator dt over denominator 1 plus straight t squared end fraction rightwards double arrow straight I space equals space 2 cross times 2 straight pi open parentheses tan to the power of negative 1 end exponent straight t close parentheses subscript 0 superscript 1 rightwards double arrow space straight I equals space 4 straight pi space tan to the power of negative 1 end exponent left parenthesis 1 right parenthesis rightwards double arrow space straight I equals 4 straight pi cross times straight pi over 4 equals straight pi squared$

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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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