Trigonometric integration formulas are essential tools in calculus. The basic integrals include sine, cosine, tangent, secant, cosecant, and cotangent functions. Each has its specific antiderivative formula that must be memorized and applied correctly.
When integrating powers of trigonometric functions, we need special techniques. For sine squared and cosine squared, we use power reduction formulas that convert them into linear combinations of constants and double angle functions. These formulas are derived from double angle identities and make integration much simpler.
Integration by parts is essential for trigonometric integrals involving products of functions. The formula u dv equals u v minus integral of v du. For example, to integrate x sine x, we choose u equals x and dv equals sine x dx. This technique is particularly useful for exponential trigonometric products and logarithmic trigonometric combinations.
Trigonometric substitution is a powerful technique for integrating expressions involving square roots. For square root of a squared minus x squared, we substitute x equals a sine theta. For square root of a squared plus x squared, we use x equals a tangent theta. For square root of x squared minus a squared, we substitute x equals a secant theta. These substitutions convert algebraic expressions into trigonometric ones that are easier to integrate.