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More U-Substitution February 17, 2009

Substitution Rule for Indefinite Integrals If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then Substitution Rule for Definite Integrals If g’(x) is continuous on [a,b] and f is continuous on the range of u = g(x), then

Indefinite Integrals by Substitution 1)Choose u. 2)Calculate du. 3)Substitute u. Arrange to have du in your integral also. (All xs and dxs must be replaced!) 4)Solve the new integral. 5)Substitute back in to get x again.

Techniques of Integration so far… 1. Use Graph & Area ( ) 2. Use Basic Integral Formulas 3. Simplify if possible (multiply out, separate fractions…) 4. Use U-Substitution…..

Choosing u Try to choose u to be an inside function. (Think chain rule.) Try to choose u so that du is in the problem, except for a constant multiple.

Choosing u For u = 3x + 2 was a good choice because (1)3x + 2 is inside the exponential. (2)The derivative is 3, which is only a constant.

Doesn’t Fit All We can’t use u–substitution to solve everything. For example: Let u = x 2 du = 2x dx We need 2x this time, not just 2. We CANNOT multiply by a variable to adjust our integral. We cannot complete this problem

Doesn’t Fit All For the same reason, we can’t do the following by u– substitution: But we already knew how to do this!

Doesn’t Fit All For the same reason, we can’t do the following by u– substitution: But we already knew how to do this!

Morals No one technique works for everything. Don’t forget things we already know! There are lots of integrals we will never learn how to solve… That’s when Simpson’s Rule, the Trapezoidal Rule, Midpoint,….. need to be used to estimate…...

Definite Integrals Evaluate:

Definite Integrals: Why change bounds? Can simplify calculations. (no need to substitute back to the original variable) Try: Change of bounds: when x = -5, u = 25 - (-5) 2 =0 when x = 5, u = 25 - (5) 2 = 0

Definite Integrals & bounds

Examples

Compare the two Integrals: Extra “x”

Notice that the extra ‘x’ is the same power as in the substitution: Extra “x”

Compare: Still have an extra “x” that can’t be related to the substitution. U-substitution cannot be used for this integral

Evaluate: Returning to the original variable “t”:

Evaluate: Returning to the original variable “t”:

Evaluate: Returning to the original variable “t”:

Use: It’s necessary to know both forms: t 2 - 2t +26 and 25 + (t-1) 2 t 2 - 2t +26 = (t 2 - 2t + 1) + (-1+26) = (t-1) 2 + 25

Completing the Square: Comes from

Use to solve: How do you know WHEN to complete the square? Ans: The equation x 2 + x + 3 has NO REAL ROOTS (Check b 2 - 4ac) If the equation has real roots, it can be factored and later we will use Partial Fractions to integrate.

Evaluate:

Even Powers of Sine

Even Powers of Cosine

Odd Powers use forms of: Save one sinx for the du If then (adjust for -) Replace the remaining even powers of sinx with sin 2 x = 1 - cos 2 x

Odd Powers use forms of: Save one cosx for the du If then Replace the remaining even powers of cosx with cos 2 x = 1 - sin 2 x

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