Introduction to Evaluation of Integrals

In a Nut Shell: The basic strategy to evaluate integrals is to transform integrals with

a complicated function into one or more basic integrals where you know each of the

basic integrals. Clearly then, you must know the basic integrals.

There are two types of integrals - - indefinite and definite. Indefinite integrals have no

limits of integration specified. Definite integrals have specified limits of integration.

Basic integrals you need to know are for the following functions.

Polynomials

Sine and cosine functions (trigonometric functions in general)

Exponential functions

Idefinite integrals (no limits of integration are specified) that you MUST know:

Terminology: the “integrand” is the function being integrated

Integral Integrand

a. ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + c xⁿ with restriction n ≠ ˗ 1

b. ∫ sin x dx = - cox x + c sin x

c. ∫ cos x dx = sin x + c cos x

d. ∫ e^ax dx = e^ax / a + c e^ax

where c is the constant of integration

Note: Differentiation of the result of the integration should return the integrand.

So you can always check to see if the result for your integration is correct.

Example: I = ∫(x + 1)² dx = ∫ [ x² + 2 x + 1 ] dx

I = x³ / 3 + x² + x + c where c is the constant of integration

Check dI/dx = x² + 2 x + 1 = (x + 1)² which yields the integrand;

So this integration was correct.

Here I is called an “indefinite” integral and c is the “constant of integration”

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