1. Determine if each integral is improper. If it is improper, state why, Rewrite it using proper limit notation, and solve if it converges-
a) -∞∫∞ 1/1+x2 dx
b) 2∫10 1/(x-2)2/3 dx
c) 0∫4 e√x/√x dx
d) e∫∞ 1/x(ln x)2 dx
e) 0∫∞ x/(x2+1)2 dx
2. Calculate the integral-
a) ∫x cox dx
b) ∫x ln√x dx
c) ∫ex sinx dx
d) ∫cos √x dx
e) 1∫e x ln 2x dx
3. Calculate the integral-
a) ∫cos3x sin4x dx
b) tan3x dx
c) tan42x dx
d) ∫(sinx sec x/sin 2x) dx
e) ∫sin3/2x cos3x dx
4. Find general solution & any particular solutions-
a) y' = 3x2(1+ y2)
b) x2y' = y - xy, y(-1) = -1
c) ln x · dy/dx = y/x
5. A certain Radioactive material is decaying at a rate proportional to amount present. If a sample of 50 g of materials was present initially, & after 2 hours, the sample lost 10% mass, Find:
a) An expression for the mass of the material remaining at any time (t).
b) Mass of material after on house.
c) Half-life of the materials.
6. A 1000-gallon tank, initially full of water developed a leak at the bottom. Given that 500 gallons of water leaked in the first 30 mins. that the water detains off a rate proportional to the amount of water present; find the amount of water left in the tank (t) minutes after the leak develops.