A definite integral calculates the area under a curve between two specific limits, resulting in a numerical value. An indefinite integral, on the other hand, finds the general antiderivative of a function, resulting in another function plus a constant of integration.
The Fundamental Theorem of Calculus states that if F(x) is an antiderivative of f(x), then the definite integral of f(x) from a to b is F(b) - F(a). Find the antiderivative, then evaluate it at the upper and lower limits of integration and subtract.
Common techniques include substitution, integration by parts, trigonometric substitution, and partial fractions. The choice of technique depends on the form of the integrand.
Choose a suitable substitution (u = g(x)), find du/dx, and rewrite the integral in terms of u. Remember to change the limits of integration to the corresponding u-values.
Use integration by parts when the integrand is a product of two functions, such as x*sin(x) or ln(x)*x. Choose u and dv strategically using the acronym LIATE (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential).
Use trigonometric identities to simplify the integrand. Common identities include sin^2(x) + cos^2(x) = 1, double angle formulas, and product-to-sum formulas. Also, consider u-substitution if applicable.
Key properties include linearity (integral of a sum is the sum of integrals), constant multiple rule, additivity (integral from a to b plus integral from b to c equals integral from a to c), and symmetry properties for even and odd functions.
Split the integral into intervals where the absolute value function is positive and negative. Rewrite the integral without the absolute value by negating the function in the intervals where its negative.
Forgetting to change the limits of integration when using substitution, incorrectly applying integration by parts, making algebraic errors during simplification, and not considering the properties of even and odd functions.