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Integrals

Short Answer Type

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

Consider the given integral
$straight I space equals integral fraction numerator left parenthesis 2 straight x minus 5 right parenthesis straight e to the power of 2 straight x end exponent over denominator left parenthesis 2 straight x minus 3 right parenthesis cubed end fraction dx Rewriting space the space above space integral space as straight I space equals space integral straight e to the power of 2 straight x minus 3 end exponent space straight x space straight e cubed fraction numerator left parenthesis 2 straight x minus 3 minus 2 right parenthesis over denominator left parenthesis 2 straight x minus 3 right parenthesis cubed end fraction dx equals space straight e cubed integral straight e to the power of 2 straight x minus 3 end exponent open square brackets fraction numerator left parenthesis 2 straight x minus 3 right parenthesis over denominator left parenthesis 2 straight x minus 3 right parenthesis cubed end fraction minus fraction numerator 2 over denominator left parenthesis 2 straight x minus 3 right parenthesis cubed end fraction close square brackets dx equals space straight e cubed integral straight e to the power of 2 straight x minus 3 end exponent open square brackets fraction numerator 1 over denominator left parenthesis 2 straight x minus 3 right parenthesis squared end fraction minus fraction numerator 2 over denominator left parenthesis 2 straight x minus 3 right parenthesis cubed end fraction close square brackets dx Let space us space consider comma space 2 straight x minus 3 space equals space straight t rightwards double arrow space 2 dx space equals space dt therefore space straight I equals straight e cubed integral straight e to the power of straight t open square brackets 1 over straight t squared minus 2 over straight t cubed close square brackets dt over 2 rightwards double arrow space straight I equals straight e cubed over 2 integral straight e to the power of straight t open square brackets fraction numerator straight t minus 2 over denominator straight t cubed end fraction close square brackets space dt Let space straight f space left parenthesis straight t right parenthesis space equals 1 over straight t squared straight f apostrophe left parenthesis straight t right parenthesis space equals space fraction numerator negative 2 over denominator straight t cubed end fraction If space straight I space equals integral straight e to the power of straight t space end exponent left square bracket straight f space left parenthesis straight t right parenthesis space plus space straight f apostrophe left parenthesis straight t right parenthesis space right square bracket dt space then comma space straight I space equals space space straight e apostrophe straight f space left parenthesis straight t right parenthesis space plus space straight C therefore space straight I space equals straight e cubed over 2 space straight x space straight e to the power of straight t space space straight x space straight f space left parenthesis straight t right parenthesis space plus space straight C equals space straight e cubed over 2 space straight x space space straight e to the power of 2 straight x minus 3 end exponent space space straight x space fraction numerator 1 over denominator left parenthesis 2 straight x minus 3 right parenthesis squared end fraction space plus space straight C equals fraction numerator straight e to the power of 2 straight x end exponent over denominator 2 left parenthesis 2 straight x minus 3 right parenthesis squared end fraction plus straight C$

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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.

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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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