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

Antiderivatives

Calculus

7. Antiderivatives & Indefinite Integrals

Antiderivatives

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Multiple Choice

Which of the following is an antiderivative of $f (x) = \cos (x^{2} - 5)$ ?

2 x \sin (x^{2} - 5) + C

0.5 \sin (x^{2} - 5) + C

\sin (x^{2} - 5) + C

- \sin (x^{2} - 5) + C

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Verified step by step guidance

Step 1: Recall that an antiderivative is a function whose derivative gives the original function. To find the antiderivative of f(x) = cos(x^2 - 5), we need to reverse the differentiation process.

Step 2: Observe that the argument of the cosine function is x^2 - 5. This suggests that the chain rule was used during differentiation. The chain rule states that if g(x) is a composite function, then its derivative is g'(x) = f'(u) * u'(x), where u is the inner function.

Step 3: Let u = x^2 - 5. Then, du/dx = 2x. This substitution simplifies the integral of f(x) = cos(x^2 - 5) into ∫cos(u) * (du/2x).

Step 4: Rewrite the integral using the substitution: ∫cos(u) * (du/2x) = (1/2) ∫cos(u) du. The antiderivative of cos(u) is sin(u), so the integral becomes (1/2) sin(u) + C.

Step 5: Substitute back u = x^2 - 5 into the result to return to the original variable: (1/2) sin(x^2 - 5) + C. This matches the correct answer provided in the problem.