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- Homework Statement
- find the derivative of ##y=x^2ln x-x## hence evaluate ##\int_1^2 xln x \ dx##

- Relevant Equations
- differentiation

##y=x^2ln x-x##

##\frac {dy}{dx}=2x ln x+x-1##

##\int [2xln x+x-1]\,dx##=##x^2ln x-x##, since ##\int -1 dx= -x##

it follows that,

##\int [2x ln x +x]\,dx##=##x^2 ln x##

##\int 2x ln x \,dx = x^2ln x##+##\int x\,dx##

##\int_1^2 xln x\,dx =\frac {x^2ln x}{2}##+##\frac{x^2}{4}##=##2ln2+1-0.25##

##\frac {dy}{dx}=2x ln x+x-1##

##\int [2xln x+x-1]\,dx##=##x^2ln x-x##, since ##\int -1 dx= -x##

it follows that,

##\int [2x ln x +x]\,dx##=##x^2 ln x##

##\int 2x ln x \,dx = x^2ln x##+##\int x\,dx##

##\int_1^2 xln x\,dx =\frac {x^2ln x}{2}##+##\frac{x^2}{4}##=##2ln2+1-0.25##

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