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Integrals

Short Answer Type

231.

space F i n d space colon integral fraction numerator left parenthesis 2 x minus 5 right parenthesis e to the power of 2 x end exponent over denominator left parenthesis 2 x minus 3 right parenthesis end fraction d x

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232.

Find space colon integral fraction numerator straight x squared plus straight x plus 1 over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction dx

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233. $Evaluate space colon space integral subscript negative 2 end subscript superscript 2 fraction numerator straight x squared over denominator 1 plus 5 to the power of straight x end fraction dx.$

Consider space the space given space integral straight I equals integral subscript negative 2 end subscript superscript 2 fraction numerator straight x squared over denominator 1 plus 5 to the power of straight x end fraction dx space space.... left parenthesis straight i right parenthesis Let space us space use space the space property comma integral subscript straight a superscript straight b straight f left parenthesis straight x right parenthesis space dx space equals integral subscript straight a superscript straight b straight f left parenthesis straight a plus straight b minus straight x right parenthesis dx therefore space straight I space equals integral subscript negative 2 end subscript superscript 2 fraction numerator left parenthesis negative straight x right parenthesis squared over denominator 1 plus 5 to the power of negative straight x end exponent end fraction dx equals integral subscript negative 2 end subscript superscript 2 fraction numerator 5 to the power of straight x space straight x squared over denominator 1 plus 5 to the power of straight x end fraction dx space.. space left parenthesis ii right parenthesis Adding space equations space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space have comma 2 straight I space equals integral subscript negative 2 end subscript superscript 2 fraction numerator 1 plus 5 to the power of straight x over denominator 1 plus 5 to the power of straight x end fraction space. space straight x squared dx equals integral subscript negative 2 end subscript superscript 2 straight x squared dx equals space open square brackets straight x cubed over 3 close square brackets subscript negative 2 end subscript superscript 2 equals space 1 third left square bracket 8 minus left parenthesis negative 8 right parenthesis right square bracket equals space 1 third space left square bracket 16 right square bracket rightwards double arrow space straight I space equals space 8 over 3

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234.

Find colon space integral left parenthesis straight x plus 3 right parenthesis square root of 3 minus 4 straight x minus straight x squared space end root space dx.

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235.

Find the integrating factor for the following differential equation: $straight x space log space straight x dy over dx space plus space straight y space equals space 2 logx space$

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236.

Evaluate: $integral subscript 0 superscript straight pi divided by 2 end superscript fraction numerator sin squared straight x over denominator sinx plus cosx end fraction dx.$

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237.

Evaluate $integral subscript negative 1 end subscript superscript 2 left parenthesis straight e to the power of 3 straight x end exponent plus 7 straight x minus 5 right parenthesis dx$

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238.

Evaluate:
$integral fraction numerator straight x squared over denominator straight x to the power of 4 plus straight x squared minus 2 end fraction dx$

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239.

Evaluate: $integral fraction numerator left parenthesis straight x plus 3 right parenthesis straight e to the power of straight x over denominator left parenthesis straight x plus 5 right parenthesis cubed end fraction dx$

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240.

Three schools X, Y, and Z organized a fete (mela) for collecting funds for flood victims in which they sold hand-held fans, mats and toys made from recycled material, the sale price of each being Rs. 25, Rs. 100 andRs. 50 respectively. The following table shows the number of articles of each type sold:
School/Article School X School Y School Z
Hand-held fans 30 40 35
Mats 12 15 20
Toys 70 55 75
Using matrices, find the funds collected by each school by selling the above articles and the total funds collected. Also write any one value generated by the above situation.

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