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Math for Econ II-Homework Assignment 6Selin KalayciogluNew York University1. Find the following integrals:(a)Z1√4-xdx(b)Z3x-2√2x+ 1dx(c)Zte5tdt(d)Z(lnt)2dt(e)Zt+ 7√5-tdt(f)Zx+ 16x+x2(g)Z1w4-w3(h)Z10x+ 2x3-5x2+x-5dx2. Iffis a twice differentiable function, findZf00(x) lnx dx+Zf(x)x2dx(Your answer should containf, but no integrals.)3. Iffis a twice differentiable function, findRxf00(x)dx.(Your answer should containf, but no integrals.)4. During a surge in the demand for electricity, the rate,r, at which energy is used can be approximated byr=te-at,wheretis the time in hours andais a positive constant.(a) Find the total energy,E, used in the firstThours. Give your answer as a function ofa.(b) What happens toEasT→ ∞?5. A rumor is spread in a school. For 0< a <1 andb >0, the timetat which a fractionpof the school populationhas heard the rumor is given byt(p) =Zpabx(1-x)dx.(a) Evaluate the integral to find an explicit formula fort(p). Write your answer so it has only one ln term.(b) At timet= 0 one percent of the school population (p= 0.01) has heard the rumor. What isa?(c) At timet= 1 half the school population (p= 0.5) has heard the rumor. What isb?(d) At what time has 90% of the school population (p= 0.9) heard the rumor?