Section 6.7. Integration by substitution
Overview
Theorem 1: Integrals of 1/x
An example
A rate of change problem
Theorem 2: Integrals of exponential functions
Finding an area
Theorem 3: Integrals of the hyperbolic cosine and sine
Theorem 4: Integrals with trigonometric functions
A velocity problem
An area problem
Theorem 5: Integrals with inverse sine and tangent:
A rate of change problem
An area
COMMENTS
Integration by substitution
Title: U-SUBSTITUTION-INDEFINITE-ANSWERS.jnt Author: mcisnero Created Date: 11/19/2011 6:52:29 PM
6 0 dx x2 + 36 Let u= x 6. Then du= dx, so Z x=6 x=0 dx x2 + 36 = 1 36 Z x=6 x=0 dx x 6 2 + 1 = 6 36 Z u=1 u=0 du u2 + 1 = 1 6 arctan(u) 1 0 = ˇ 24: 4.Evaluate the following inde nite integral. This can be done with only one substitution, but may be easier to approach with two. Hint: Use u= x2 for the rst substitution, rewrite the integral in ...
6.1 integration by substitution
Integration by Substitution SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference Chapter 5.3 of the rec-ommended textbook (or the equivalent chapter in your alternative textbook/online resource) and your lecture notes. EXPECTED SKILLS: Know how to simplify a \complicated integral" to a known form by ...
INTEGRATION BY SUBSTITUTION Page 1 of 5. Basic integration. In its most basic form, using the Fundamental Theorem of Calculus, an indefinite integral is simply Z f (x) dx = F (x) + C, where is an arbitrary constant and F (x) is 0the antiderivative of f ), that = f ). If the integral is definite, Z. a. 2. a. 1. f (x) dx = F (a. 2) F (a. 1) .
Integration by Substitution Homework - Free download as PDF File (.pdf), Text File (.txt) or read online for free. Lesson 6.7: Integration by Substitution
Then work the Interactive Examples to practice the concepts and techniques before you start the Exercises. TOPICS IN THIS SECTION. 1 Overview. 2 Theorem 1: Integrals of 1/x. 3 An example. 4 A rate of change problem. 5 Theorem 2: Integrals of exponential functions. 6 Finding an area. 7 Theorem 3: Integrals of the hyperbolic cosine and sine.
INTEGRATION by substitution
Activity 5.3.3. Evaluate each of the following indefinite integrals by using these steps: Find two functions within the integrand that form (up to a possible missing constant) a function-derivative pair; Make a substitution and convert the integral to one involving u u. and du; d u; Evaluate the new integral in u; u;
Substitution for Definite Integrals. Express each definite integral in terms of u, but do not evaluate. Evaluate each definite integral. ∫ −12 x2 ( 4 x3 − 1)3 dx; u = 4 x3 − 1.
a good choice for a u -substitution is making substituüon into the integral, we have Therefore, I by of 3. (1 pt) alfredLibrary/AUCWcha terMesonWdeEniteusub7.pg Consider the definite integral (2x — dz. TtEn the most substitution to simpliW this making tl'E substitution, changing tl'E limits of integra- tion, arKl simpliWing, we obtain 4.
3 Z 10x. [Area] =. dx. 0 (x2 + 1)2. We evaluate this definite integral by making a change of variables in the corresponding indefinite integral. We use the substitution u = x2 + 1, for which. du =. (x2 + 1) dx = 2x dx. To construct du from x dx, we write 10 = 5(2) and dx put the 2 with the x and dx to obtain.
Title: U-SUBSTITUTION-Def. Integrals- ANSWERS.jnt Author: mcisnero Created Date: 11/19/2011 7:30:24 PM
We will calculate the integral following the steps outlined. (1) Choose a substitution u= g(x). Observing that, were we to calculate the derivative of the integrand, we would eventually need to use the chain rule to di erentiate tan(2x+ 1) and need to write y= tanu; u= 2x+ 1, we choose the substitution u= 2x+ 1.
Today: Integration by substitution. Homework before Tuesday's class: watch video 9.4, as well as 9.5, 9.6. Boris Khesin MAT137 January 27, 2022 1 / 13. Warm up Calculate Z sin p x p x dx Hint: Use the substitution u = p x. Boris Khesin MAT137 January 27, 2022 2 / 13 Let ffHd×=FC )+c. then /
Integration by Substitution SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference Chapter 5.3 of the rec- ... 6. Z (3x 5)9 dx 7. Z e 2x dx 8. Z sinxcosx 1+sin2 x dx 9. Z csc x 2 cot x 2 dx 10. Z 33x p 1 x4 dx 11. Z e p x p x 1 4 cos4x dx 1. 12. Z 1 2+4x2 dx 13. Z 4x (3+x2)2 dx 14. Z x2 p 4 xdx. 15. Z ...
Example 4.3.1. Determine the general antiderivative of. h(x) = (5x − 3)6. Check the result by differentiating. For this composite function, the outer function f is f(u) = u6, while the inner function is u(x) = 5x − 3. Since the antiderivative of f is F(u) = 1 7u7 + C, we see that the antiderivative of h is.
7) ∫36 x3(3x 4 + 3)5 dx; u = 3x4 + 3 8) ∫x(4x − 1) dx; u = 4x − 1 -1- ©L f2v0 S1z3 U NKYu1tPa 1 TS9o3f Vt7w UazrpeT CL pLbCG.T T 7A fl Ylw driTg Nh0tns U JrQeVsje Br 1vIe cd g.p g rM KaLdzeG fw riEtGhK lI 3ncf XiKn8iytZe0 9C5aYlBc Ru1lru 8si.p Worksheet by Kuta Software LLC
Activity 5.3.3. Evaluate each of the following indefinite integrals by using these steps: Find two functions within the integrand that form (up to a possible missing constant) a function-derivative pair; Make a substitution and convert the integral to one involving u and du; Evaluate the new integral in u;
Integration by Substitution SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference Chapter 5.3 of the rec-ommended textbook (or the equivalent chapter in your alternative textbook/online resource) and your lecture notes. EXPECTED SKILLS: Know how to simplify a \complicated integral" to a known form by ...