Integration: This is defined as anti- differentiation. Suppose, y = x3 + 2x, the first derivative is
3x2 + 2. (dy/dx = 3x2 + 2)
INTEGRATION BY ALGEBRAIC SUBSTITUTION
Sometimes integral are not given in the standard form, such integral are then reduced to standard form format before evaluation by algebraic substitution.
Suppose, an integral is given in the form ∫ f(ax + b)n dx
Then, the algebraic substitution is to represent the function in the bracket by any letter.
Evaluate the following integrals:
1. ∫(5x – 7 )7/2dx 2. ∫cos 9x dx 3. ∫ xcos 2x dx.
INTEGRATION BY PARTS
This technique is uniquely useful in evaluating integrals that are not in the standard form. Such integrals cannot be solved by algebraic substitution.
From the product rule of differentiation, it can be generalized thus;
∫ vdu = uv – ∫ udv.
INTEGRATION BY PARTIAL FRACTION
Sometimes rational functions are not expressed in the proper standard form; such function can be evaluated by transforming them into standard form through partial fractions. The knowledge of partial fractions is needed here to evaluate the functions.