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: $\int_{}^{} 7 x^{2} + 49 d x$ .

\frac{7}{3} x^{3} + 49 x + C

7 x^{3} + 49 x + C

7 x^{2} + 49 x + C

\frac{7}{2} x^{2} + 49 x + C

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Verified step by step guidance

Step 1: Recognize that the problem involves finding the indefinite integral of the function $ 7x^2 + 49 $ with respect to $ x $. Recall that the indefinite integral represents the antiderivative of the function.

Step 2: Break the integral into separate terms for easier computation: $ \int (7x^2 + 49) \, dx = \int 7x^2 \, dx + \int 49 \, dx $. This uses the property of integrals that $ \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx $.

Step 3: Apply the power rule for integration to $ \int 7x^2 \, dx $. The power rule states that $ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C $, where $ n \neq -1 $. For $ 7x^2 $, the coefficient 7 remains unchanged, and the exponent increases by 1: $ \int 7x^2 \, dx = 7 \cdot \frac{x^{2+1}}{2+1} = \frac{7x^3}{3} $.

Step 4: Integrate the constant term $ \int 49 \, dx $. Recall that the integral of a constant $ c $ is $ cx + C $. Therefore, $ \int 49 \, dx = 49x $.

Step 5: Combine the results from Step 3 and Step 4 to write the complete antiderivative: $ \int (7x^2 + 49) \, dx = \frac{7x^3}{3} + 49x + C $, where $ C $ is the constant of integration.