Evaluate the integral, if it exists.\[\int_1^\infty \frac{4}{x^3} \;dx\]

Suppose the cost of utilities for a house is given by \[C(t) = 50 \cos\left(\frac{\pi}{6}t\right) + 100,\] where \(t\) is in months. Find the average cost of utilities from December (\(t=0\)) to June (\(t=6\)).

Find the equation of the tangent line to the curve \[y = 3 \arctan(2x)\] at the point \(\displaystyle x = \frac{1}{2}\).

Order: Omeprazole 11mg, gastrostomy tube, every 12 hours, for reflux  Child’s weight: 22 kg Safe dose range: 1-1.5mg/kg/day in two divided doses   a) What is the safe low dose (mg/dose) for this child? b) What is the safe high dose (mg/dose) for this child?  c) Is the ordered dose safe for this child?

Find the total mass of the \(1\) m rod whose linear density function is given by \[\rho(x) = 4x(1+x^2)^3 \;\;\text{    kg/m}\] for \(0 \leq x \leq 1\).

Which of the following is the partial fractions decomposition of the rational function \[f(x)=\frac{5x^2+1}{(x^2-1)^2(x^2+9)}\;\;?\]

Order: Vancomycin 600mg, intravenously, every 6 hours, for moderate infection  Child’s weight: 50 kg  Safe dose range: 10-15mg/kg/dose (max dose 750mg) every 6 hours    a) What is the safe low dose (mg/dose) for this child?  b) What is the safe high dose (mg/dose) for this child? c) Is the ordered dose safe for this child?  

Evaluate the indefinite integral \[\int_1^e  \ln\left(x^2\right)\;dx.\] (Hint: Integration by Parts.)

Evaluate the integral \[\int_0^{\pi/2} \cos^3{x}\;dx.\](Hint: \(\sin^2{x}+\cos^2{x}=1\))

Order: Morphine 0.5mg, intravenously, every 4 hours, for pain  Patient’s weight: 6.31 kg  Safe dose range: 0.05-0.1mg/kg/dose every 4 hours    a) What is the safe low dose (mg/dose) for this child? Round to the nearest hundredth.  b) What is the safe high dose (mg/dose) for this child? Round to the nearest hundredth.   c) Is the ordered dose safe for this child?