What is the result of evaluating the integral ∫ x^3 dx?

What is the result of evaluating the integral ∫ x^3 dx?

• A. (1/3)x^3 + C
• B. (1/4)x^4 + C
• C. (1/5)x^5 + C
• D. (1/2)x^2 + C

Answer: (B)

To evaluate the integral ∫ x^3 dx, we can apply the power rule of integration. The power rule states that the integral of x^n, where n is a constant, is given by: ∫ x^n dx = (1/(n+1)) x^(n+1) + C In this case, n is 3. Therefore, we can calculate the integral as follows: 1. Add 1 to the exponent: n + 1 = 3 + 1 = 4. 2. Divide by the new exponent: (1/4) x^(4). 3. Don’t forget to add the constant of integration, C. This results in: ∫ x^3 dx = (1/4) x^4 + C. Thus, the correct outcome of evaluating the integral is indeed (1/4)x^4 + C. This aligns perfectly with the proper application of the power rule, confirming the choice. The confusion regarding other options comes from potentially misapplying the power rule for different exponents, but the step-by-step process leads directly to the selected answer.

To evaluate the integral ∫ x^3 dx, we can apply the power rule of integration. The power rule states that the integral of x^n, where n is a constant, is given by:

∫ x^n dx = (1/(n+1)) x^(n+1) + C

In this case, n is 3. Therefore, we can calculate the integral as follows:

1. Add 1 to the exponent: n + 1 = 3 + 1 = 4.

2. Divide by the new exponent: (1/4) x^(4).

3. Don’t forget to add the constant of integration, C.

This results in:

∫ x^3 dx = (1/4) x^4 + C.

Thus, the correct outcome of evaluating the integral is indeed (1/4)x^4 + C. This aligns perfectly with the proper application of the power rule, confirming the choice. The confusion regarding other options comes from potentially misapplying the power rule for different exponents, but the step-by-step process leads directly to the selected answer.