Integral Rules; Integration by Substitution

Integral Rules; Integration by Substitution Rules for indefinite integration 1. Constant Multiple Rule : (Note k is outside of the function) 2. Rule for Negatives : (Rule 1 with k = 1) 3. Sum and Difference Rule :

Integral Rules; Integration by Substitution The Power and Chain Rule in Integral Form When u is a differentiable function of x, and n is a rational number (n  1) , the Chain Rule tells us that This same equation , says that u n+1/(n+1) is one of the antiderivatives of the function u n(d u/ dx) . Therefore ,

Integral Rules; Integration by Substitution From the previous slide: The integral on the left-hand side of this equation is usually written in the simpler “differential” form , obtained by treating the dx’s as differentials that cancel. To summarize, If u is any differentiable function , then Remember that we are thinking of the u’s as functions of x

Example 1: One of the clues that we look for is if we can find a function and its derivative in the integrand. The derivative of is . Note that this only worked because of the 2x in the original. Many integrals can not be done by substitution.

If we let u = g(x) then du = g’(x)dx Substitution : Running the Chain Rule backward If we let u = g(x) then du = g’(x)dx So the integral becomes Since the F(u) is the antiderivative of f(u) Then we get this when we substitute g(x) back in Notice the use of the small “f” and the big “F” The method works because F(g(x)) is an antiderivative of f(g(x)) • g′(x) whenever F is an antiderivative of f :

The main idea of integration using substitution In each case in the previous example and in the examples that follow, we can use integration by substitution only because we have a composite function which has a function on the inside for which we have a derivative on the outside.

Example 2: We solve for because we can see it in the integrand.

Example 3: This one is tricky because there does not seem to be anything on the outside, but since the derivative of the inside is a constant, it does not matter . Solve for dx.

Example 4:

Remember that sin 4 (x) means (sin x)4 Example 5: Remember that sin 4 (x) means (sin x)4

EXAMPLE 6 Evaluate.

SOLUTION

EXAMPLE 7 Evaluate.

SOLUTION

EXAMPLE 8 Evaluate.

SOLUTION

EXAMPLE 9 Evaluate.

SOLUTION…

…SOLUTION

EXAMPLE 10 Evaluate.

SOLUTION

Taking the derivative !!! Remember that we can always check any integration problem by ???? Taking the derivative !!!