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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.

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

straight I space equals integral left parenthesis straight x plus 3 right parenthesis square root of 3 minus 4 straight x minus straight x squared end root dx Let space left parenthesis straight x space plus 3 right parenthesis space equals space straight A straight d over dx left parenthesis 3 minus 4 straight x minus straight x squared right parenthesis space plus space straight B space space space... space left parenthesis straight i right parenthesis rightwards double arrow space space straight x space plus space 3 space equals space straight A left parenthesis negative 2 straight x minus 4 right parenthesis space plus space straight B rightwards double arrow space straight x plus 3 space equals space minus 2 Ax minus 4 straight A plus straight B therefore space minus 2 straight A space space equals space 1 rightwards double arrow space straight A space equals space minus 1 half minus 4 straight A space plus space straight B space equals space 3 rightwards double arrow space minus 4 space open parentheses negative 1 half close parentheses plus space straight B space equals 3 rightwards double arrow space space straight B equals 1 therefore comma straight I space equals integral open square brackets negative 1 half straight d over dx left parenthesis 3 minus 4 straight x minus straight x squared right parenthesis plus 1 close square brackets space square root of 3 minus 4 straight x minus straight x squared end root space dx equals space space 1 half integral straight d over dx left parenthesis 3 minus 4 straight x minus straight x squared right parenthesis square root of 3 minus 4 straight x minus straight x squared end root dx space plus space integral square root of 3 minus 4 straight x minus straight x squared minus 4 plus 4 dx end root equals space minus 1 half fraction numerator left parenthesis 3 minus 4 straight x minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent over denominator 3 divided by 2 end fraction space plus space integral square root of 7 minus left parenthesis straight x plus 2 right parenthesis squared end root dx equals space minus fraction numerator left parenthesis 3 minus 4 straight x minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent over denominator 3 end fraction space plus space fraction numerator straight x plus 2 over denominator 2 end fraction square root of 7 minus left parenthesis straight x plus 2 right parenthesis squared dx end root equals space minus fraction numerator left parenthesis 3 minus 4 straight x minus straight x squared right parenthesis to the power of begin display style 3 over 2 end style end exponent over denominator 3 end fraction space plus space fraction numerator straight x plus 2 over denominator 2 end fraction square root of 7 minus left parenthesis straight x plus 2 right parenthesis squared end root space plus space 7 over 2 sin to the power of negative 1 end exponent open parentheses fraction numerator straight x plus 2 over denominator square root of 7 end fraction close parentheses space space plus space straight C

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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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