Substitution and change of variables
One of the integration techniques that is useful in evaluating indefinite integrals that do not seem to fit the basic formulas is substitution and change of variables. This technique is often compared to the chain rule for differentiation because they both apply to composite functions. In this method, the inside function of the composition is usually replaced by a single variable (often u). Note that the derivative or a constant multiple of the derivative of the inside function must be a factor of the integrand.
The purpose in using the substitution technique is to rewrite the integration problem in terms of the new variable so that one or more of the basic integration formulas can then be applied. Although this approach may seem like more work initially, it will eventually make the indefinite integral much easier to evaluate.
Note that for the final answer to make sense, it must be written in terms of the original variable of integration.
Example 6: Evaluate
Because the inside function of the composition is x 3 + 1, substitute with
Because the inside function of the composition is 5 x, substitute with
Example 8: Evaluate
Because the inside function of the composition is 9 – x 2, substitute with
Integration by parts
Another integration technique to consider in evaluating indefinite integrals that do not fit the basic formulas is integration by parts. You may consider this method when the integrand is a single transcendental function or a product of an algebraic function and a transcendental function. The basic formula for integration by parts is
where u and v are differential functions of the variable of integration.
A general rule of thumb to follow is to first choose dv as the most complicated part of the integrand that can be easily integrated to find v. The u function will be the remaining part of the integrand that will be differentiated to find du. The goal of this technique is to find an integral, ∫ v du, which is easier to evaluate than the original integral.
Example 9: Evaluate ∫ x sec 2 x dx.
Example 10: Evaluate ∫ x 4 In x dx.
Example 11: Evaluate ∫ arctan x dx.
Integrals involving powers of the trigonometric functions must often be manipulated to get them into a form in which the basic integration formulas can be applied. It is extremely important for you to be familiar with the basic trigonometric identities, because you often used these to rewrite the integrand in a more workable form. As in integration by parts, the goal is to find an integral that is easier to evaluate than the original integral.
Example 12: Evaluate ∫ cos 3 x sin 4 x dx
Example 13: Evaluate ∫ sec 6 x dx
Example 14: Evaluate ∫ sin 4 x dx
If an integrand contains a radical expression of the form a specific trigonometric substitution may be helpful in evaluating the indefinite integral. Some general rules to follow are
1. If the integrand contains
2. If the integrand contains
3. If the integrand contains
Right triangles may be used in each of the three preceding cases to determine the expression for any of the six trigonometric functions that appear in the evaluation of the indefinite integral.
Example 15: Evaluate
Because the radical has the form