Let h(x) be an antiderivative of the function f(x)=x3+−sin⁡xx2+2. Which of the following is a correct expression for h(x)?

1 4 x 4 − cos ⁡ x x 2 + 2 x + C

Let h ( x ) be an antiderivative of the function f ( x ) = x 3 + − sin ⁡ x x 2 + 2 . Which of the following is a correct expression for h ( x ) ?

1 4 x 4 − cos ⁡ x x 2 + 2 x + C

1 4 x 4 + sin ⁡ x x 2 + 2 x + 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
- 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

Let $h (x)$ be an antiderivative of the function $f (x) = x^{3} + \frac{- \sin x}{x^{2}} + 2$ . Which of the following is a correct expression for $h (x)$ ?

\frac{1}{3} x^{3} + \cos x + 2 x^{2} + C

x^{4} + \sin x + 2 x + C

\frac{1}{4} x^{4} - \frac{\cos x}{x^{2}} + 2 x + C

\frac{1}{4} x^{4} + \frac{\sin x}{x^{2}} + 2 x + C

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Verified step by step guidance

Step 1: Recall that an antiderivative of a function f(x) is a function h(x) such that the derivative of h(x) equals f(x). This means we need to integrate the given function f(x) = x^3 + \frac{\sin x}{x^2} + 2 to find h(x).

Step 2: Break the function f(x) into separate terms for easier integration: f(x) = x^3 + \frac{\sin x}{x^2} + 2. We will integrate each term individually.

Step 3: For the first term, x^3, the integral is \int x^3 dx = \frac{1}{4}x^4. This follows from the power rule of integration: \int x^n dx = \frac{x^{n+1}}{n+1} for n ≠ -1.

Step 4: For the second term, \frac{\sin x}{x^2}, rewrite it as \sin x \cdot x^{-2}. The integral of this term is more complex and involves substitution. The result is -\frac{\cos x}{x}.

Step 5: For the third term, 2, the integral is \int 2 dx = 2x. Finally, combine all the terms and add the constant of integration C to get h(x) = \frac{1}{4}x^4 - \frac{\cos x}{x} + 2x + C.