to evaluate (integral) 3x^2 cos (2x^3-4) dx, it is necessary to let

A. u=3x^2
B. u=6x
C. u=2x^3-4
D. u=6x^2

To evaluate the integral ∫ 3x^2 cos(2x^3 - 4) dx, we can use the technique of substitution. Let's go through the options provided and determine which one is the best choice for the substitution.

A. u = 3x^2
If we choose u = 3x^2, it would lead to du = 6x dx, which allows us to partially simplify the integral. However, this substitution does not cover the entire expression inside the cosine function.

B. u = 6x
If we choose u = 6x, it would lead to du = 6 dx, but this substitution does not capture the x^2 term in the original integral.

C. u = 2x^3 - 4
If we choose u = 2x^3 - 4, it leads to du = 6x^2 dx. This substitution covers the entire expression inside the cosine function and also includes the x^2 term of the original integral.

D. u = 6x^2
If we choose u = 6x^2, it leads to du = 12x dx, but this does not cover the entire expression inside the cosine function.

From the options provided, the best choice for the substitution is C. u = 2x^3 - 4.

By substituting u = 2x^3 - 4, the integral becomes ∫ cos(u) du, which is a simpler integral to evaluate.

Please note that while the procedure for evaluating the integral is described here, actually solving the integral is beyond the capabilities of this text-based format.

C

then du = 2(3x^2) dx