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: $\int_{}^{} \frac{cos (x) - 1}{x} d x$ .

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

+ C

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

+ C

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

+ C

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

+ C

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Verified step by step guidance

Step 1: Recognize that the integral involves the term (cos(x) - 1)/x, which suggests using the Taylor series expansion for cos(x). Recall that cos(x) can be expressed as a power series: cos(x) = ∑_{n=0}^{∞} (-1)^n x^{2n} / (2n)!.

Step 2: Subtract 1 from the Taylor series expansion of cos(x). This modifies the series to exclude the constant term (n=0), leaving: cos(x) - 1 = ∑_{n=1}^{∞} (-1)^n x^{2n} / (2n)!.

Step 3: Divide the resulting series (cos(x) - 1) by x. This adjusts the powers of x in the series, resulting in: (cos(x) - 1)/x = ∑_{n=1}^{∞} (-1)^n x^{2n-1} / (2n)!.

Step 4: Integrate term-by-term. The integral of each term in the series ∫ x^{2n-1} dx is x^{2n}/(2n), and the factorial term (2n)! remains unchanged. This gives: ∫ (cos(x) - 1)/x dx = ∑_{n=1}^{∞} (-1)^n x^{2n} / [2n (2n)!] + C.

Step 5: Verify the series representation and compare it to the given answer choices. Ensure that the derived series matches one of the provided options, and note that the constant of integration (C) is added to account for the indefinite integral.