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

Taking space the space given space function straight I equals 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 Let space fraction numerator straight x squared space plus space straight x space plus space 1 over denominator left parenthesis straight x squared space plus space 1 right parenthesis space left parenthesis straight x space plus space 2 right parenthesis end fraction space space equals space fraction numerator straight A over denominator straight x space plus space 2 end fraction space plus space fraction numerator Bx space plus space straight C over denominator straight x squared plus 1 end fraction equals space fraction numerator straight A left parenthesis straight x squared space plus space 1 right parenthesis space plus space left parenthesis Bx space space plus space straight C right parenthesis left parenthesis straight x space plus space 2 right parenthesis over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction equals fraction numerator left parenthesis straight A plus straight B right parenthesis straight x squared plus left parenthesis 2 straight B plus straight C right parenthesis space straight x plus left parenthesis 2 straight C plus straight A right parenthesis over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x plus 2 right parenthesis end fraction Thus space equating space the space coefficients comma space we space have comma straight A space plus space straight B space equals 1 space.. space left parenthesis straight i right parenthesis 2 straight B space space plus space straight C space equals space 1 space.. left parenthesis ii right parenthesis 2 straight C space plus space straight A space equals space 1 space.. space left parenthesis iii right parenthesis solving space the space above space three space equations comma space we space have comma straight A space equals 3 over 5 comma space straight B space equals 2 over 5 space and space straight C equals 1 fifth therefore comma 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 space fraction numerator straight A over denominator straight x plus 2 end fraction plus fraction numerator Bx plus straight C over denominator straight x squared plus 1 end fraction rightwards double arrow space 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 space equals space fraction numerator 3 over denominator 5 left parenthesis straight x plus 2 right parenthesis end fraction space plus space fraction numerator 2 straight x plus 1 over denominator 5 left parenthesis straight x squared plus 1 right parenthesis end fraction therefore comma straight I equals 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 equals integral open square brackets fraction numerator 3 over denominator 5 space left parenthesis straight x plus 2 right parenthesis end fraction plus fraction numerator 2 straight x plus 1 over denominator 5 left parenthesis straight x squared plus 1 right parenthesis end fraction dx close square brackets equals 3 over 5 integral fraction numerator dx over denominator left parenthesis straight x plus 2 right parenthesis end fraction dx space plus space 1 fifth integral fraction numerator 2 straight x plus 1 over denominator left parenthesis straight x squared plus 1 right parenthesis end fraction dx equals 3 over 5 log space left parenthesis straight x plus 2 right parenthesis space plus 1 fifth integral fraction numerator 2 straight x over denominator left parenthesis straight x squared plus 1 right parenthesis end fraction dx space plus 1 fifth integral fraction numerator dx over denominator left parenthesis straight x squared plus 1 right parenthesis end fraction equals 3 over 5 log space left parenthesis straight x plus 2 right parenthesis space plus 1 fifth log space left parenthesis straight x squared plus 1 right parenthesis plus 1 fifth tan to the power of negative 1 end exponent space straight x space plus space straight C

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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.

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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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