2. Use the above table to transform the integral into a new integral problem using the integration by parts formula below:

Transcription

1 Integration by Parts: Basic ( ) Due: Mon Aug :1 AM MDT Question Instructions Read today's tes and Learning Goals 1. Question Details sp16 by parts intro 1 [34458] Find the indefinite integral (antiderivative) of 2x e x dx 1. Split the integral into two factors: u = 2x and dv = e x dx Then find the derivative of u and the antiderivative of dv. du = derivative of u multiplied by dx (du = u'dx) v = antiderivative of dv (do not include "+C") u = 2x v = du = dv = e x dx 2. Use the above table to transform the integral into a new integral problem using the integration by parts formula below: te: At this point you have not found the antiderivative, you have just transformed the integral into a new integral problem which may be easier to work with. Some problems may ask to stop here and this is a required step when showing work. 3. Find the antiderivative of the new integral problem. (te, you only need to find the antiderivative of the second piece and make sure to include a constant of integration, +C.) 2x e x dx = 4. Check the answer by taking its derivative and seeing if you can simplify it to the original integrand.

2 2. Question Details sp16 by parts intro 2 kmod [344512] Find the indefinite integral (antiderivative) of 12t sin(2t) dt 1. Split the integral into two factors: u = 12t and dv = sin(2t)dt Then find the derivative of u and the antiderivative of dv. du = derivative of u multiplied by dt (du = u'dt) v = antiderivative of dv (do not include "+C") u = 12t v = du = dv = sin(2t)dt 2. Use the above table to transform the integral into a new integral problem using the integration by parts formula below: te: At this point you have not found the antiderivative, you have just transformed the integral into a new integral problem which may be easier to work with. Some problems may ask to stop here and this is a required step when showing work. 3. Find the antiderivative of the new integral problem. (te, you only need to find the antiderivative of the second piece and make sure to include a constant of integration, +C.) 12t sin(2t) dt = 4. Check the answer by taking its derivative and seeing if you can simplify it to the original integrand.

3 3. Question Details sp15 by parts intro 3 [326245] Find the indefinite integral (antiderivative) of t 2 ln(t) dt 1. Split the integral into two factors: u = ln(t) and dv = t 2 dt Then find the derivative of u and the antiderivative of dv. du = derivative of u multiplied by dx (du = u'dx) v = antiderivative of dv (do not include "+C") u = ln(t) v = du = dv = t 2 dt 2. Use the above table to transform the integral into a new integral problem using the integration by parts formula below: te: At this point you have not found the antiderivative, you have just transformed the integral into a new integral problem which may be easier to work with. Some problems may ask to stop here and this is a required step when showing work. 3. Find the antiderivative of the new integral problem. (te, you only need to find the antiderivative of the second piece and make sure to include a constant of integration, +C.) t 2 ln(t) dt = 4. Check the answer by taking its derivative and seeing if you can simplify it to the original integrand.

4 4. Question Details by parts walk through short 1 [315476] Find the indefinite integral (antiderivative) (2x + 7) e 2x dx 1. Let u = 2x + 7 and dv = e 2x dx and use Integration By Parts to transform the integral = uv = u dv v du te: At this stage you have only transformed the integral into a new integral problem which may be easier to work with. Some problems may ask to stop here and this is a required step when showing work. 2. Find the antiderivative of the new integral problem. (te, you only need to find the antiderivative of the second piece and make sure to include a constant of integration, +C.) (2x + 7) e 2x dx = You should check the answer by taking its derivative. 5. Question Details by parts with bounds 1 [315696] Find the definite integral π 3t sin(t) dt 1. Let u = 3t and dv = sin(t) dt and use Integration By Parts to transform the integral (Evaluate (enter in the bounds) for the first piece and enter in the exact value.) π π π 2. Evaluate the new integral problem and use it to solve the original problem. (Enter in the exact answer.) 3t sin(t) dt π = π

5 6. Question Details by parts walk through twice [ ] Find the indefinite integral (antiderivative) 6x 2 e 3x dx twice. 1. Apply integration by parts with u = 6x 2 and dv = e 3x dx 2. Apply integration by parts a second time. Use u = 4x and dv = e 3x dx. 3. Finish the problem. 6x 2 e 3x dx = 7. Question Details by parts with bounds 4 [315768] Find the definite integral 1. Let u = 18x 2 and dv = e 3x dx and use Integration By Parts to transform the integral (Evaluate (enter in the bounds) for the first piece and enter in the exact value.) 2. Evaluate the new integral problem and use it to solve the original problem. (Hint: Let 1 18x 2 e 3x dx u = 12x and dv = e 3x dx (Enter in the exact answer.) 18x 2 e 3x dx = 1 and use integration by parts on the second piece.) 1

6 8. Question Details by parts walk through short 6 [315491] Find the indefinite integral (antiderivative) ln(x) dx 1. Let u = ln(x) and dv = dx and use Integration By Parts to transform the integral = uv = u dv v du 2. Find the antiderivative of the new integral problem. ln(x) dx = You should check the answer by taking its derivative.

7 9. Question Details sp16 by parts guided [ ] Suppose you wanted to transform the integral x 2 ln(x) dx 1. Answer the following / questions: Can x 2 be easily differentiated? Is x 2 a possible choice for u? Can ln(x) be easily differentiated? Is ln(x) a possible choice for u? Can x 2 be easily antidifferentiated? Is x 2 dx a possible choice for dv? Can ln(x) be easily antidifferentiated? Is ln(x) dx a possible choice for dv? 2. Based on these responses, what would be a good choice for u and dv? u = dv = 3. Using your choice of u and dv, transform the above integral. You do not need to evaluate it. x 2 ln(x) dx =

8 1. Question Details sp16 by parts guided 2 [ ] Suppose you wanted to transform the integral (4x+3) e 2x dx 1. Answer the following / questions: Can (4x+3) be easily antidifferentiated? Is (4x+3) dx a possible choice for dv? Can e 2x be easily antidifferentiated? Is e 2x dx a possible choice for dv? Based on these responses, it may not be clear what would be a good choice for u and dv. 2. Answer these further / questions: Does differentiating (4x+3) result in a "simpler" function? Does differentiating e 2x result in a "simpler" function? 3. Based on these responses, what would be a good choice for u and dv? u = dv = 4. Using your choice of u and dv, transform the above integral. You do not need to evaluate it. (4x+3) e 2x dx =

9 11. Question Details sp15 by parts MC 1 [326291] If you wanted to transform the integral (5x 2)e.1x dx using integration by parts, what would be a good choice for u and dv? Warning! This problem is penalty scored. Don't guess. u = (5x 2) dx and dv = e.1x u = 5x 2 and dv = e.1x u = e.1x dx and dv = (5x 2) u = e.1x and dv = (5x 2) dx u = 5x 2 and dv = e.1x dx Using your choice of u and dv rewrite the above integral (5x 2)e.1x = 12. Question Details sp15 by parts MC 2 [ ] If you wanted to transform the integral 7x 3 ln(x) dx using integration by parts, what would be a good choice for u and dv? Warning! This problem is penalty scored. Don't guess. u = 7x 3 and dv = ln(x) dx u = ln(x) and dv = 7x 3 dx u = ln(x) dx and dv = 7x 3 u = 7x 3 dx and dv = ln(x) Using your choice of u and dv rewrite the above integral 7x 3 ln(x) dx = 13. Question Details fa15 e2 review by parts [ ] Use Integration by Parts to evaluate the following integral. Include +C. 24xcos(4x) dx = Practice another version. Try to get three correct in a row. Assignment Details