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.
Practise AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
