Question Details sp15 u sub intro 1(MEV) [ ]

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1 12 Basic: Integration by Substitution ( ) Question Instructions Read today's Notes and Learning Goals 1. Question Details sp15 u sub intro 1(MEV) [ ] Suppose you need to compute this antiderivative: 4x 3 (x ) 7 dx The method of substitution, often called usubstitution, transforms this problem into a problem whose antiderivative is known. In this problem the method begins with this equation: u = x Step 1. Write the differential of u, namely du. This is the derivative of u(x), multiplied by dx, namely du = u'(x)dx. You must include the differential of x (which is dx) when you type your answer. du = Step 2. Substitute, with these two moves: Replace (x ) 7 with u 7. Replace the rest of the original problem with du. The result is a new antiderivative problem written with the variable u. Enter the new integrand and differential below. Step 3. Compute the transformed antiderivative. Write your result with the variable u. Include + C in your response. Step 4. Replace u in your result with x 4 +10, to convert back to the original variable x. This step completes the evaluation of the original problem. Write your result with the variable x. Include + C. 4x 3 (x ) 7 dx = Note: You should personally verify the correctness of your final result by writing its derivative. Doing so will reproduce the original integrand and provide insight into why this method works.

2 2. Question Details sp15 u sub intro 2(MEV) [ ] Follow the steps to transform this antiderivative problem into a new antiderivative problem with the method of substitution, using u = sin(x). sin 3 (x)cos(x) dx Step 1. Use the equation for u to write du in terms of x and dx. du = Step 2. Substitute. Replace sin 3 (x) with u 3. Replace the rest with du. Enter the new integrand and differential below. Step 3. Compute the transformed antiderivative. Write your result with the variable u. Include + C in your response. Step 4. Replace u in your result with sin(x), to convert back to the original variable x. This step completes the evaluation of the original problem. Write your result with the variable x. Include + C. sin 3 (x)cos(x) dx = 3. Question Details u sub 4a du mod (MEV) [ ] Transform this antiderivative problem using the substitution u = x 2. 2x e x2 dx Enter the new integrand and differential below, written entirely in terms of u. Note: Some problems, like this one, will only request the transformed antiderivative problem. They will not require you to actually find an antiderivative, nor will they require you to convert back to the original variable. 4. Question Details u sub 4a du mod 2(MEV) [ ] Transform this antiderivative problem using the substitution u = ln(x). ln(x) x dx Enter the new integrand and differential below, written entirely in terms of u.

3 5. Question Details u sub 4a du mod 3(MEV) [ ] The method of substitution can be more complex. This problem uses the substitution problem u = 2x 3 4 in the antiderivative x 2 (2x 3 4) 9 dx Note that du = 6x 2 dx, so that x 2 dx = Now enter the new integrand and differential, written entirely with u. 6. Question Details u sub 4a du mod 4(MEV) [ ] Transform this antiderivative problem using the substitution u = x 2 4. x dx x 2 4 Note that du = 2x dx, so that x dx = Enter the new integrand and differential, written entirely with u. Compute the antiderivative, in terms of u. Include + C. Convert back to the original variable, to complete the original problem. Write in terms of x. Include + C. x dx x 2 4 =

4 7. Question Details u sub 4a du mod2 (MEV) [ ] Transform this antiderivative problem using the substitution u = cos(x). tan(x) dx = sin(x) cos(x) dx The substitution implies that sin(x) dx= Write the new antiderivative problem, in terms of u. Compute the antiderivative, in terms of u. Include + C. Complete the original problem. Write in terms of x. Include + C. tan(x) dx = 8. Question Details sp15 antiderivative review 3 mod(mev) [ ] Linear arguments (namely, the form a+bx, as seen below) are a special case because the derivative of the linear argument is the constant b. Substitution can be used, but it is useful to know how to address this special case without using usubstitution. You will not be required to show work for such problems and can consider them to be elementary antiderivatives. If the argument is not linear, so that its derivative is not constant, then you cannot use this shorter method. Compute this antiderivative. Then you can practice with more problems that are similar. (Use + C for the constant of integration.) 3 9x + 15 dx = Practice another version to get more examples.

5 9. Question Details sp15 u sub intro 2 (MEV) [ ] The method of substitution is also useful for definite integrals. The substitution equation provides new integration limits, after which converting back to the original variable is optional. Consider using 2 x 2 (2x 3 4) 9 dx 1 u = 2x 3 4 to transform As before, du = 6x 2 dx, so that x 2 dx = Write the transformed problem with u a b Use the substitution equation, first with x=1 and then with x=2, to compute new limits of integration. These limits are now values of u. a = u(1)= b = u(2)= Note: This integral is now written entirely in terms of u. Some problems may ask you to stop at this point. Other problems may require you to compute the integral with the Fundamental Theorem of Calculus (FTC). You have two options, and the choice is yours. One option is to compute with u and the new limits of integration, as you have now written above. The other option is to convert back to the original variable x, as you did in previous problems, and to use the original limits of integration. Use your chosen option to evaluate the transformed problem using the FTC. Write your answer as an exact value. You do not need to simplify your answer. Do not use decimals.

6 10. Question Details sp15 u sub intro 2 3(MEV) [ ] Use the substitution u = sin(x) to transform and then compute the integral π /6 sin 4 (x)cos(x) dx 0 The substitution gives the differential du = and the new lower and upper integration limits a = u(0) = b = u π = 6 Write the newly transformed integral, in terms of u. π /6 sin 4 (x)cos(x) dx = 0 a b Compute the integral, using the FTC. Use the variable u and its integration limits, as written above, or convert back to the original variable x and the original integration limits, as you prefer. Enter your result in the box, as an exact value. Do not use decimals.

7 11. Question Details sp15 u sub intro 2 4(MEV) [ ] Use the substitution u = π t 2 2 cos π t dt 1 2 to transform and then compute the integral The substitution gives the differential du = from which dt = The substitution equation also gives the new integration limits a = u(1)= b = u(2)= Write the transformed integral, in terms of u. 2 cos π t dt 1 2 = a b Compute the integral, using the FTC. Enter your result in the box, as an exact value. Do not use decimals. 12. Question Details fa15 usub random versions 1 [ ] Transform this antiderivative problem using the substitution u = 1 + e x. e x (1 + e x ) 3 dx Your result should be a new antiderivative problem written in terms of u. Antidifferentiate in terms of u, then convert back to x to complete the original problem. Include + C in your response. e x (1 + e x ) 3 Practice another version. This problem will generate many different examples in which substitution is appropriate. You will need to practice another version many times to see all the different possibilities. dx =

8 13. Question Details SP14 UMC 1 MEV [ ] If you wanted to transform the antiderivative problem (x 2 + 2)(x 3 + 6x) 5 dx using a usubstitution, which of the following would be the best choice for u? u = 3x u = x u = 5(x 3 + 6x) 4 (3x 2 + 6) u = x 3 + 6x 14. Question Details SP14 UMC 2 MEV [ ] If you wanted to transform the antiderivative problem 3te t2 +2 dt using a usubstitution, which of the following would be the best choice for u? u = 2t u = e t2 u = 3t u = t Question Details SP14 USA 3 MEV [ ] If you wanted to transform the antiderivative problem cos 4 x sin x dx via substitution, what is the best choice for u? u = Use your choice to transform the problem. Write your answer in terms of u:

9 16. Question Details SP14 USA 4 MEV [ ] If you wanted to transform the antiderivative problem (ln x) 2 x dx via substitution, what is the best choice for u? u = Use your choice to transform the problem. Write your answer in terms of u: 17. Question Details SP14 USA 1 MEV [ ] If you wanted to transform the integral 4 (12x + 10) 3x 2 + 5x dx 1 via substitution, what is the best choice for u? u = Use this choice to transform the integral. Your final answer should be a new integral written in terms of u: b in which a a = and b = Assignment Details Name (AID): 12 Basic: Integration by Substitution ( ) Submissions Allowed: 100 Category: Homework Code: Locked: Yes Author: Skriletz, Jaimos ( jaimosskriletz@boisestate.edu ) Last Saved: Dec 10, :30 PM MST Group: BSU Calculus Randomization: Person Which graded: Last Feedback Settings Before due date Question Score Assignment Score Publish Essay Scores Question Part Score Mark Add Practice Button Help/Hints Response Save Work After due date Question Score Assignment Score Publish Essay Scores Key Question Part Score Solution Mark Add Practice Button Help/Hints Response

Question Details SCalcET [ ] Question Details SCalcET [ ] Question Details SCalcET

Question Details SCalcET [ ] Question Details SCalcET [ ] Question Details SCalcET Vector Products I (11812924) Due: Fri Jan 12 2018 01:30 PM MST Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Instructions Notes and Learning Goals Things to think about: The dot and cross products