Use the substitution u=1+tu = 1 + \sqrt{t}u=1+t to show that the integral
∫12t1+t dt \int \frac{12\sqrt{t}}{1+\sqrt{t}} \, dt ∫1+t12tdtcan be written in the form
∫(24u−48+24u) du \int \left( 24u - 48 + \frac{24}{u} \right) \, du ∫(24u−48+u24)duThe mass of a fungal colony, mmm grams, grows at a rate modelled by the equation
dmdt=12t1+t \frac{dm}{dt} = \frac{12\sqrt{t}}{1+\sqrt{t}} dtdm=1+t12twhere ttt is the number of days since the colony was first observed, for 1≤t≤91 \le t \le 91≤t≤9.
Determine the total increase in the mass of the colony from the end of day 1 to the end of day 9. Show each stage of your working and give your answer to one decimal place.
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.
