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Answer :
Final answer:
To integrate the function 66cos^3(40x)dx, we use substitution and trigonometric identities, resulting in a process that simplifies the trigonometric function for straightforward integration. The correct option is B.
Explanation:
To integrate the function 66cos3(40x), we can employ substitution and trigonometric identities. First, note that cos3(θ) can be expressed using the trigonometric identity cos3(θ) = (3cos(θ) + cos(3θ))/4. Applying this to our integral, we substitute u = 40x, which leads to du = 40dx, and thus dx = du/40. The integral then becomes 66/40 ∗ integral of (3cos(u) + cos(3u))/4 du. This can be further simplified and integrated directly to obtain a function in terms of u, and finally substituting back to get the function in terms of x. This is a straightforward use of substitution and trigonometric identities to simplify and then integrate a trigonometric function.
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