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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.

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235.
Find the integrating factor for the following differential equation: $straight x space log space straight x dy over dx space plus space straight y space equals space 2 logx space$

Consider the given differential equation:
$xlogx dy over dx plus straight y space equals space 2 logx$
Dividing the above equation by xlogx, we have,
$space space fraction numerator xlogx space over denominator xlogx end fraction dy over dx space plus space fraction numerator straight y over denominator straight x space logx end fraction space equals space fraction numerator 2 logx over denominator xlogx end fraction rightwards double arrow dy over dx plus Py space equals space straight Q comma space where space straight P space and space straight Q space are space functions space of space straight x Comparing space equation space left parenthesis 1 right parenthesis space and space the space general space equation comma space we space have comma space straight P left parenthesis straight x right parenthesis space equals space fraction numerator 1 over denominator straight x space logx end fraction space and space straight Q left parenthesis straight x right parenthesis space equals space 2 over straight x The space intergrating space factor space is space given space by space the space formulae space straight e to the power of integral Pdx end exponent space Thus comma space straight I. straight F. space equals space straight e to the power of integral Pdx end exponent space space equals space straight e to the power of integral fraction numerator dx over denominator straight x space log space straight x end fraction end exponent$
$Consider space straight I space equals space integral dx over xlogx Substituting space logx equals space straight t semicolon space space dx over straight x space equals space dt Thus space straight I space equals space integral dt over straight t space equals space log left parenthesis straight t right parenthesis space equals space log space left parenthesis log space straight x right parenthesis Hence comma space straight I. straight F. equals space straight e to the power of integral fraction numerator dx over denominator straight x space log space straight x end fraction end exponent space equals space straight e to the power of log left parenthesis log space straight x right parenthesis end exponent space equals space log space straight x$

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

459 Views

239.

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

519 Views

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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Previous Year Papers

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Test Yourself

Short Answer Type

235. Find the integrating factor for the following differential equation:

235.
Find the integrating factor for the following differential equation: $straight x space log space straight x dy over dx space plus space straight y space equals space 2 logx space$