Topics in Integration

Calculating the definite integral of a given real function f (x), $$\int_a^b {f(x)dx,} $$ is a classic problem. For some simple integrands f (x), the indefinite integral $$\int_a^x {f\left( x \right)} dx = F\left( x \right),F'\left( x \right) = f\left( x \right),$$ can be obtained in closed form as an algebraic expression in x and wellknown transcendental functions of x. Then $$\int_a^b {f(x)dx = F(b) - F(a).} $$ See Gröbner and Hofreiter (1961) for a comprehensive collection of formulas describing such indefinite integrals and many important definite integrals.