Determining the Limits Equal to the Integral of 31sin(x^3+2) dx and Matching with (1/3)[-cos(x^3+2)] | from a to b

Which of the following limits is equal to ∫31sin(x3+2)ⅆx ?

To determine which of the given limits is equal to the integral of 31sin(x^3+2) dx, let’s evaluate the integral first:

∫31sin(x^3+2) dx

To find the anti-derivative of sin(x^3+2), we can make a substitution u = x^3+2

To determine which of the given limits is equal to the integral of 31sin(x^3+2) dx, let’s evaluate the integral first:

∫31sin(x^3+2) dx

To find the anti-derivative of sin(x^3+2), we can make a substitution u = x^3+2. Then, du = 3x^2 dx, giving us dx = du / (3x^2).

Now, substituting the values:

∫31sin(x^3+2) dx = ∫31sin(u) (du / (3x^2))

Since the integral is in terms of u, we need to find the limits of integration in terms of u as well. Given that the original limits are not provided, we’ll assume them to be a and b.

So, the integral becomes:

∫31sin(u) (du / (3x^2))

Now, we can integrate sin(u):

(1/3) ∫31sin(u) du

Using the antiderivative of sin(x) as -cos(x):

(1/3) [-cos(u)] | from a to b

Now, substituting back u = x^3+2:

(1/3) [-cos(x^3+2)] | from a to b

Therefore, the integral of 31sin(x^3+2) dx is (1/3) [-cos(x^3+2)] | from a to b.

From the given options, we need to determine which limit matches this result. Since the limits are not specified, we can’t directly compare them. However, by inputting different values for a and b into the limits, we can rearrange the terms to match the form (1/3) [-cos(x^3+2)] | from a to b.

Once the limits are matched, we can identify the correct option.

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