Use the substitution u=2+sinθu = 2 + \sin \thetau=2+sinθ to show that the integral
∫24sin2θ(2+sinθ)3dθ \int \frac{24 \sin 2\theta}{(2 + \sin \theta)^3} d\theta ∫(2+sinθ)324sin2θdθcan be written in the form
∫(48u2−96u3)du \int \left( \frac{48}{u^2} - \frac{96}{u^3} \right) du ∫(u248−u396)duThe torque τ\tauτ (in N m) generated by a mechanical component is modeled by the function
τ(θ)=24sin2θ(2+sinθ)3,0≤θ≤π2 \tau(\theta) = \frac{24 \sin 2\theta}{(2 + \sin \theta)^3}, \quad 0 \le \theta \le \frac{\pi}{2} τ(θ)=(2+sinθ)324sin2θ,0≤θ≤2πwhere θ\thetaθ is the angle of rotation in radians. Calculate the exact value of the total work done, given by ∫0π/2τ(θ)dθ\int_{0}^{\pi/2} \tau(\theta) d\theta∫0π/2τ(θ)dθ.
Show each stage of your working and give your answer as a fraction in its simplest form.
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.
