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MATH 121 Lecture Notes - Lecture 21: Natural Logarithm, Power Rule, Antiderivative

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brownsardine137

9 Apr 2017

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First let"s remember how we took the derivative using the chain rule. It is not as easy as it seems to be. It is sometimes helpful to use a process called substitution. Therefore, integration by substitution is related to the chain rule. Change a more complex function into a simple function of u, including the differential du. (cid:1516)(cid:1858)(cid:4666)(cid:1876)(cid:4667)(cid:1856)(cid:1876) (cid:1516)(cid:1858)(cid:4666)(cid:1873)(cid:4667)(cid:1856)(cid:1873). Note that: (cid:1858)(cid:4666)(cid:1873)(cid:4667) and (cid:1856)(cid:1873) need to have the same variable. In general, for the types of problems we are concerned with, there are three cases. We choose u to be one of the following: the quantity under a root or raised to a power: (base) n x dx. Note: some integrands may need to be rearranged to fit one of these cases. Find the antiderivative for the following. (remember we want to change the more complex function into a single variable like u. ) Example 4: (decide which type of the following functions)

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Related Questions

(1) Integration by substitution is related to what differentiation methord?

(A) Integration by substitution is related to the exponential rule and logarithm rule.

(B) Integration by substitution is related to the sum or difference rule,

(C) Integration by substitution is related to the chain rule.

(D) Integration by substitution is related to the power rule and quotient rule.

(2) What type of inegrand suggests using integration by substitution?

(A) Integrands without constants.

(B) Integrands with a root, a power a denominator, or a factorial.

(C) Intergrands that are the product or quetient of two functions.

(D) Integrands with a root, a power, a denominator, or an exponent on e.

need help on 2 and 3 please

Trigonometric Substitution Rules

For v(a^2-u^2 ) let u = asin?? and v(a^2-u^2 ) = acos??
For v(a^2+ u^2 ) let u = atan? and v(a^2+u^2 ) = asec?
For v(u^2-a^2 ) let u = asec?? and v(u^2-a^2 ) = atan?

1. Integrate ?dx/v(25+4x^2 ). Identify the following:

a = u = du =

What adjustment, if any, needs to be made to the integral tocomplete the substitution in to u?

2. Rewrite the above integral so that it is all in u.

3. Which of the trig substitution rules above will we use to solvethis integral? Why? Draw and label the relevant right trianglediagram.

4. Use the rule you chose and the diagram to identify what you willreplace u, du and v(a^2+u^2 ) with.

5. Rewrite the integral from step 2 using the replacements fromstep 4. Did you use all the replacements? If not, why not? Can thisintegral be simplified? If so, do it?

6. Integrate the resulting trig function using a basicformula.

7. Use the information from steps 1 and 4 to rewrite the answer instep 6 in terms of the variable x.

u = 2x = 5tan? and v(a^2+u^2 )=v(25+4x^2 )=5sec?

8. How does knowing the trigonometric relationships as defined bythe three sides of a right triangle and the Pythagorean Theoremhelp us solve integrals using trig substitutions?

9. ?dx/(xv(49-9x^2 )) 10. ?dx/v(16x^2-1)

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