Evaluate the indefinite integral as a power series: ∫11−t7 dt.

∑ n = 0 ∞ t 7 n + 1 7 n + 1 + C

Evaluate the indefinite integral as a power series: ∫ 1 1 − t 7 d t .

∑ n = 0 ∞ t 7 n + 1 7 n + 1 + C

∑ n = 0 ∞ t n + 1 n + 1 + C

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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
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10. Physics Applications of Integrals 3h 16m
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  1h 13m
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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
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- Taylor Series & Taylor Polynomials
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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 a power series: $\int \frac{1}{1 - t^{7}} d t$ .

\sum_{n = 0}^{\infty} \frac{t^{7 n + 1}}{7 n + 1}

+ C

\sum_{n = 0}^{\infty} \frac{t^{n + 1}}{n + 1}

+ C

\sum_{n = 0}^{\infty} t^{7 n}

+ C

\sum_{n = 1}^{\infty} \frac{t^{n}}{n}

+ C

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Verified step by step guidance

Step 1: Recognize that the integrand $ \frac{1}{1 - t^7} $ resembles the geometric series formula $ \frac{1}{1 - x} = \sum_{n=0}^{\infty} x^n $, which is valid for $ |x| < 1 $. Rewrite $ \frac{1}{1 - t^7} $ as $ \sum_{n=0}^{\infty} t^{7n} $.

Step 2: Substitute the series representation $ \sum_{n=0}^{\infty} t^{7n} $ into the integral $ \int \frac{1}{1 - t^7} \, dt $, resulting in $ \int \sum_{n=0}^{\infty} t^{7n} \, dt $.

Step 3: Use the property of integration to integrate term-by-term for power series. The integral of $ t^{7n} $ is $ \frac{t^{7n+1}}{7n+1} $, so the series becomes $ \sum_{n=0}^{\infty} \frac{t^{7n+1}}{7n+1} $.

Step 4: Add the constant of integration $ C $ to account for the indefinite integral. The result is $ \sum_{n=0}^{\infty} \frac{t^{7n+1}}{7n+1} + C $.

Step 5: Verify the convergence of the series for $ |t| < 1 $, ensuring the validity of the geometric series expansion used in the solution.