Reference no: EM131106131

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) ∫e^x 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) ∫sin^3/2x 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.