Determine ∫sin2θ1+sinθ dθ\int \frac{\sin 2\theta}{1 + \sin \theta} \, d\theta∫1+sinθsin2θdθ
A research team monitors the rate of energy consumption, EEE (in gigajoules per hour), of a large-scale data center. The rate is modeled by the function f(t)=18t2t+3f(t) = \frac{18t}{\sqrt{2t + 3}}f(t)=2t+318t, where ttt is the time in hours since midnight.
Use the substitution u=2t+3u = \sqrt{2t + 3}u=2t+3 to show that
∫18t2t+3 dt=2(2t+3)12(At+B)+k \int \frac{18t}{\sqrt{2t + 3}} \, dt = 2(2t + 3)^{\frac{1}{2}}(At + B) + k ∫2t+318tdt=2(2t+3)21(At+B)+kwhere AAA and BBB are integers to be found and kkk is an arbitrary constant.
