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1.8 Integration

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Question 69
i.

Find

∫e2xex+1 dx \int \frac{e^{2x}}{\sqrt{e^x + 1}} \, dx ∫ex+1​e2x​dx
[5]
ii.

During a chemical reaction, the rate of change of the mass mmm of a byproduct, in grams per hour, is modeled by the equation

dmdt=27t3t+4,t≥0 \frac{dm}{dt} = \frac{27t}{\sqrt{3t + 4}}, \quad t \ge 0 dtdm​=3t+4​27t​,t≥0

Use the substitution u=3t+4u = \sqrt{3t + 4}u=3t+4​ to show that

∫27t3t+4 dt=2(3t+4)12(At+B)+k \int \frac{27t}{\sqrt{3t + 4}} \, dt = 2(3t + 4)^{\frac{1}{2}}(At + B) + k ∫3t+4​27t​dt=2(3t+4)21​(At+B)+k

where AAA and BBB are integers to be found and kkk is a constant of integration.

[7]
Markscheme

1.8 Integration Questions

  1. A Level
  2. /Maths
  3. /1.8 Integration

438 exam-style questions on OCR A Level Maths 1.8 Integration, covering 1.8.1 Fundamental theorem of calculus (A-level only), 1.8.2 Integrating x^n, 1.8.3 Integrating standard functions (A-level only), 1.8.4 Evaluating definite integrals, 1.8.5 Area between a curve and the x-axis, 1.8.6 Area between two curves, 1.8.7 Integration as the limit of a sum (A-level only), 1.8.8 Integration by substitution (A-level only), 1.8.9 Integration by parts (A-level only), 1.8.10 Use of partial fractions in integration (A-level only), 1.8.11 Differential equations with separable variables (A-level only), 1.8.12 Interpreting the solution of a differential equation (A-level only), and 1.8 Integration. Each one has a worked solution and a mark scheme showing where the marks go.

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