Using the substitution u=3x+1u = \sqrt{3x+1}u=3x+1, show that
∫1512x+4e3x+1 dx \int_{1}^{5} \sqrt{12x+4} e^{\sqrt{3x+1}} \, dx ∫1512x+4e3x+1dxmay be expressed in the form
∫abku2eu du \int_{a}^{b} k u^2 e^u \, du ∫abku2euduwhere aaa, bbb and kkk are constants to be found.
Hence find, by algebraic integration, the exact value of
∫1512x+4e3x+1 dx \int_{1}^{5} \sqrt{12x+4} e^{\sqrt{3x+1}} \, dx ∫1512x+4e3x+1dxgiving your answer in simplest form.
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.