Maths (university) · Part 06
Integration techniques and uses
Harder integrals, volumes, simple differential equations and numerical methods.
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- Integration by partsThe integral of u dv equals uv minus the integral of v du. It is the product rule run backwards, used when the integrand is a product of two different kinds of function.
- Choosing u and dvA good choice makes the new integral simpler than the old one. Logarithms and inverse trig functions go in u. Exponentials and sines are usually dv.
- Partial fractionsSplit a fraction with a factorised denominator into simpler fractions that are easy to integrate.
- Trigonometric integralsUse identities to turn powers of sine and cosine into forms you can integrate. For example, sin^2 x = (1 - cos 2x)/2.
- Improper integralAn integral with an infinite limit, or a function that blows up inside the interval. Replace the problem point with a variable and take a limit.
- Volume of revolutionSpin a region around the x-axis. Slice it into thin discs of radius f(x), each with volume pi f(x)^2 dx, and add them up with an integral.
- Numerical integrationWhen no antiderivative can be found, estimate the integral with the trapezoid rule or Simpson's rule, which use values of the function at several points.
- Separable differential equationAn equation for dy/dx that can be rearranged to have all the y terms on one side and all the x terms on the other. Then integrate both sides.
- Work as an integralWhen a force changes with position, work is the integral of force over distance. For a spring, force is kx, where k is the spring constant.