Evaluate ∫ x^4 dx.

Evaluate ∫ x^4 dx.

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

Answer: (A)

To evaluate the integral ∫ x^4 dx, we apply the power rule for integration. According to this rule, when integrating a term of the form x^n, where n is any real number, the integral is given by: ∫ x^n dx = (1/(n+1)) x^(n+1) + C, where C is the constant of integration. In this case, n is 4. Thus, we will calculate: 1. Increase the exponent by 1, which gives us n + 1 = 4 + 1 = 5. 2. Divide by this new exponent (5) to get (1/5). 3. Multiply by x raised to the new exponent: (1/5)x^(5). Putting it all together, the result of the integral is: ∫ x^4 dx = (1/5)x^5 + C. This confirms that the correct choice is indeed (1/5)x^5 + C. The other given choices represent integrals of different functions and thus do not align with the integral of x^4.

To evaluate the integral ∫ x^4 dx, we apply the power rule for integration. According to this rule, when integrating a term of the form x^n, where n is any real number, the integral is given by:

∫ x^n dx = (1/(n+1)) x^(n+1) + C,

where C is the constant of integration.

In this case, n is 4. Thus, we will calculate:

1. Increase the exponent by 1, which gives us n + 1 = 4 + 1 = 5.

2. Divide by this new exponent (5) to get (1/5).

3. Multiply by x raised to the new exponent: (1/5)x^(5).

Putting it all together, the result of the integral is:

∫ x^4 dx = (1/5)x^5 + C.

This confirms that the correct choice is indeed (1/5)x^5 + C. The other given choices represent integrals of different functions and thus do not align with the integral of x^4.