Integrals and Differential Equations

About this unit

Integration reverses differentiation. In this unit you first find antiderivatives (indefinite integrals) using standard forms, then learn the three main methods for harder integrands: substitution, partial fractions and integration by parts. Definite integrals, evaluated through the fundamental theorem of calculus, give exact areas under curves. The unit ends with differential equations, which are equations involving derivatives; you learn their order and degree and solve the common first-order types. These ideas model growth, decay, motion and many economic situations.

Key ideas

• ∫ f(x) dx = F(x) + C whenever F′(x) = f(x).
• Substitution, partial fractions and integration by parts handle non-standard integrands.
• ∫ from a to b of f(x) dx = F(b) − F(a).
• Properties such as ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx simplify tricky definite integrals.
• Area under y = f(x) between x = a and x = b is a definite integral.
• Order and degree describe a differential equation; variable separable, homogeneous and linear equations have standard methods.

How to prepare

Learn the table of standard integrals as well as you know the derivative table. Practise choosing the right method quickly. For areas, always sketch the region first. For differential equations, first identify the type, then apply its method step by step.