How Do You Evaluate the Indefinite Integral?

How Do You Evaluate the Indefinite Integral?

In general, different types of indefinite integrals require different strategies, and not all indefinite integrals can be solved in a step-by-step manner. The most basic method of finding an indefinite integral is to rewrite it and then use a table of integrals.

  1. Rewrite the integral as a sum of simpler integrals

    If the integral involves the sum of two or more functions, use the additive property of the integral to write it as a sum of integrals. For instance, the integral of 3 sin(x) + x can be rewritten as the integral of 3 sin(x) plus the integral of x.

  2. Pull out any multiplicative constants

    If any of the functions to integrate are multiplied by numbers, use the constant multiplicative property of the integral to write the multiplication outside the integral. An example is that the integral of 3 sin(x) can be written as 3 times the integral of sin(x).

  3. Look up the antiderivatives in a table

    At this point, the integral should be written as a sum of several integrals. Look up each integral in a table; tables of integrals are readily available online and in calculus textbooks. Replace each integral with its entry in the table. Don't forget to write the "+C." Memorizing the entries in the table of integrals makes this process go more quickly.

  4. Use other methods, if necessary

    If one or more of the integrals can't be found in the table, try using a more advanced method such as Integration By Parts or Trigonometric Substitution. If these methods still don't work, computer software may be used to find the integral.