Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

12. Techniques of Integration

Integration by Parts

Calculus

12. Techniques of Integration

Integration by Parts

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Multiple Choice

For the integral $\int_{}^{} x e^{x} d x$ , which of the following correctly identifies $u$ and $d u$ for use in integration by parts?

u = e^{x}, d u = d x

u = x, d u = e^{x} d x

u = x, d u = d x

u = e^{x}, d u = e^{x} d x

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Verified step by step guidance

Step 1: Recall the formula for integration by parts: $ \int u \, dv = uv - \int v \, du $. This formula is used to break down integrals involving products of functions.

Step 2: Identify the two components of the integrand $ x e^{x} $. Typically, we choose $ u $ to be the part that simplifies when differentiated, and $ dv $ to be the part that is easy to integrate.

Step 3: Choose $ u = x $ because differentiating $ x $ simplifies it to $ du = dx $. Let $ dv = e^{x} dx $, which integrates to $ v = e^{x} $.

Step 4: Substitute $ u $, $ du $, $ v $, and $ dv $ into the integration by parts formula: $ \int x e^{x} \, dx = uv - \int v \, du $. This becomes $ x e^{x} - \int e^{x} \, dx $.

Step 5: Simplify the remaining integral $ \int e^{x} \, dx $, which is straightforward since the integral of $ e^{x} $ is $ e^{x} $. Combine the results to express the solution.