Find ∫cosx dx\displaystyle\int \cos x\,dx∫cosxdx.
You may use the result ddx(tankx)=ksec2kx\dfrac{d}{dx}(\tan kx) = k\sec^2 kxdxd(tankx)=ksec2kx. Find ∫2sec24x dx\displaystyle\int 2\sec^2 4x\,dx∫2sec24xdx.
Hence, using the identity tan2θ≡sec2θ−1\tan^2\theta \equiv \sec^2\theta - 1tan2θ≡sec2θ−1, find ∫tan2θ dθ\displaystyle\int \tan^2\theta\,d\theta∫tan2θdθ.
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.