# Take the following indefinite integrals a 5 points z

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6. Take the following indefinite integrals.(a) (5 points)Zln(x)xdx.(hint: u-substitution)Lettingu= ln(x)gives usdudx=1x, ordu=1xdx.Zln(x)xdx=Zu du=u22+C=(ln(x))22+C(b) (5 points)Z2x2-xdx.(hint: partial fractions)
7. Solve the following initial value problem.
8. (10 points) You’re drinking coffee while working on a practice final, but you’re soconcentrated on the questions that you forget to continue drinking.Your coffeestarts at150F and the temperature of the room is75F. According to Newton’s Lawof Cooling, the coffee’s temperaturey(t)(in degrees Fahrenheit) as a function of time(in minutes) is given by the differential equationdydt=75-y50y(0) = 150,How long until the coffee reaches100F?This is a separable differential equation. It can be rewritten as1(75-y)dy=150dt,and integrating both sides gives us-ln(75-y) =t50+C.We now solve foryin terms oft.ln(75-y) =-t50-C75-y=e-t50-C75-y=e-t50e-C(lete-C=C0)75-y=C0e-t50y(t) = 75-C0e-t50,The initial condition isy(0) = 150. Plugging this in we get150 =y(0) = 75-C0e-050= 75-C0=C0=-75.Therefore the temperature of the coffee isy(t) = 75+75e-t/50. To answer the questionof how long it takes for the coffee to reach100F, we write the equationy(t) = 100and solve fort.75 + 75e-t/50= 10075e-t/50= 25e-t/50=13-t50= ln(13)t=-50 ln(13)(23.86minutes)
TABLE OFINTEGRALSBASICFUNCTIONS1.Zxndx=1n+ 1xn+1+C(ifn6
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