Differentials and integrals: Integration techniques
Introduction
In examples of direct integration we have already applied two calculation rules for integrating:
Constant factor rule A constant factor can be taken outside the integral: \[\int c\cdot f(x)\,\dd x=c\cdot \int f(x)\,\dd x.\]
Sum rule, the integral of the sum of two functions is the sum of the two integrals: \[\int \bigl(f(x)+g(x)\bigr)\,\dd x=\int f(x)\,\dd x+\int g(x)\,\dd x.\]
These two rules for integration correspond to two calculation rules for differentiating. We look at more translations of calculation rules for differention into techniques to calculate primitive functions.
It is not true that a primitive function in the form of a mathematical formula, , in terms of standard functions (powers, exponential functions, logarithms, trigonometric functions, etc.), exists for any mathematical function and can be found. One of the highlights of the integral calculus is the Risch algorithm which decides whether a function has an integral in an elementary formula form and if so, what the mathematical formula then exactly is. This mathematical algorithm is very complicated and is not based on the techniques that are discussed in this section.
