tegration by Parts titution or Inte- -Parts for- SOCA 2012 3. Why is u = cos x, du = x d. Ixcos 15. x cos(x) dx, 13. x² sin cosx dx? fet sin 7.) /e-51 8 −5* sin x d.x 21. fx²in.xdx ln 2 sin x dx 19. fx ln xdx sin x dx
tegration by Parts titution or Inte- -Parts for- SOCA 2012 3. Why is u = cos x, du = x d. Ixcos 15. x cos(x) dx, 13. x² sin cosx dx? fet sin 7.) /e-51 8 −5* sin x d.x 21. fx²in.xdx ln 2 sin x dx 19. fx ln xdx sin x dx
tegration by Parts titution or Inte- -Parts for- SOCA 2012 3. Why is u = cos x, du = x d. Ixcos 15. x cos(x) dx, 13. x² sin cosx dx? fet sin 7.) /e-51 8 −5* sin x d.x 21. fx²in.xdx ln 2 sin x dx 19. fx ln xdx sin x dx
Can you help me solve problem 17 using integration by parts?
Thank you!
Transcribed Image Text:### Integration by Parts Practice Problems
Here are some practice problems that require using the method of integration by parts. Try solving each integral and compare your answers.
---
1. **Evaluate the following integrals:**
13. \[\int x^2 \sin x \, dx\]
15. \[\int e^{-x} \sin x \, dx\]
17. \[\int e^{-5x} \sin x \, dx\]
19. \[\int x \ln x \, dx\]
21. \[\int x^2 \ln x \, dx\]
---
### Tips for Integration by Parts
1. Using the integration by parts formula:
\[
\int u \, dv = uv - \int v \, du
\]
Identify parts of the integrand to set as \( u \) and \( dv \).
2. Choose \( u \) and \( dv \) such that differentiating \( u \) (to get \( du \)) and integrating \( dv \) (to get \( v \)) simplifies the integral.
---
### Example Steps for Solving Problem 13:
1. \[\int x^2 \sin x \, dx\]
- Let \( u = x^2 \) and \( dv = \sin x \, dx \).
- Therefore, \( du = 2x \, dx \) and \( v = -\cos x \).
- Apply the integration by parts formula:
\[
\int x^2 \sin x \, dx = -x^2 \cos x + \int 2x \cos x \, dx
\]
Repeat integration by parts on the remaining integral if necessary.
### Practice each of these problems and compare your solutions with examples provided in your textbooks or online resources to confirm accuracy.
This list of integrals provides a broad range of straightforward to complex problems which will help in mastering the technique of integration by parts.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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![### Integration by Parts Practice Problems
Here are some practice problems that require using the method of integration by parts. Try solving each integral and compare your answers.
---
1. **Evaluate the following integrals:**
13. \[\int x^2 \sin x \, dx\]
15. \[\int e^{-x} \sin x \, dx\]
17. \[\int e^{-5x} \sin x \, dx\]
19. \[\int x \ln x \, dx\]
21. \[\int x^2 \ln x \, dx\]
---
### Tips for Integration by Parts
1. Using the integration by parts formula:
\[
\int u \, dv = uv - \int v \, du
\]
Identify parts of the integrand to set as \( u \) and \( dv \).
2. Choose \( u \) and \( dv \) such that differentiating \( u \) (to get \( du \)) and integrating \( dv \) (to get \( v \)) simplifies the integral.
---
### Example Steps for Solving Problem 13:
1. \[\int x^2 \sin x \, dx\]
- Let \( u = x^2 \) and \( dv = \sin x \, dx \).
- Therefore, \( du = 2x \, dx \) and \( v = -\cos x \).
- Apply the integration by parts formula:
\[
\int x^2 \sin x \, dx = -x^2 \cos x + \int 2x \cos x \, dx
\]
Repeat integration by parts on the remaining integral if necessary.
### Practice each of these problems and compare your solutions with examples provided in your textbooks or online resources to confirm accuracy.
This list of integrals provides a broad range of straightforward to complex problems which will help in mastering the technique of integration by parts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd3f7c47f-f942-4013-9970-37881347efaf%2F8f06af2c-693f-4049-892a-e7eb7ac0ec02%2Fuvgz75_processed.jpeg&w=3840&q=75)
