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)324sin2θdθ
can be written in the form
∫(48u2−96u3)du\int \left( \frac{48}{u^2} - \frac{96}{u^3} \right) du∫(u248−u396)du
The 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≤2π
where 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 Edexcel A Level Maths 11.5 Integration by Substitution with exam-style questions for A Level Maths. 59 questions, matched to the Edexcel A Level Maths (9MA0) 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.