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

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Calculus

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

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Multiple Choice

Evaluate the indefinite integral as an infinite series: $\int_{}^{} (\cos (x) - 1) x d x$ .

\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{2 n + 2}}{(2 n + 2) (2 n + 1)!}

+ C

\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n} x^{2 n + 1}}{(2 n + 1)!}

+ C

\sum_{n = 1}^{\infty} \frac{x^{2 n}}{(2 n)!}

+ C

\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} x^{2 n + 1}}{(2 n + 1) (2 n)!}

+ C

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Verified step by step guidance

Step 1: Begin by understanding the problem. The goal is to evaluate the indefinite integral ∫ (cos(x) - 1) x dx as an infinite series. This involves expanding the function cos(x) into its Taylor series representation and integrating term by term.

Step 2: Recall the Taylor series expansion for cos(x): cos(x) = ∑_{n=0}^{∞} (-1)^n x^(2n) / (2n)!. Subtract 1 from this series to get cos(x) - 1 = ∑_{n=1}^{∞} (-1)^n x^(2n) / (2n)! (note that the n=0 term vanishes).

Step 3: Multiply the series representation of cos(x) - 1 by x. This results in x(cos(x) - 1) = ∑_{n=1}^{∞} (-1)^n x^(2n+1) / (2n)!.

Step 4: Integrate term by term. The integral of x^(2n+1) is x^(2n+2) / (2n+2). Therefore, the integral becomes ∑_{n=1}^{∞} (-1)^n x^(2n+2) / [(2n+2)(2n)!] + C, where C is the constant of integration.

Step 5: Simplify the series expression to match the given answer format. The final result is ∑_{n=1}^{∞} (-1)^n x^(2n+1) / [(2n+1)(2n)!] + C, which is the correct answer.