Which of the following is a step in evaluating.
(Integral) cos^2 5x dx
A. (Integral) 1+cos10x/2 dx
B. (Integral) 1-cos10x/2 dx
C. (Integral) 1+cos10x/20 dx
D. (Integral) 1-cos10x/20 dx
I'd say A, since cos10x = 2cos^2 5x - 1
To evaluate the integral ∫ cos^2(5x) dx, we can use the trigonometric identity:
cos^2(x) = (1 + cos(2x))/2.
Using this identity, we can transform the integral as follows:
∫ cos^2(5x) dx = ∫ (1 + cos(10x))/2 dx.
So, the answer is option A: ∫ (1 + cos(10x))/2 dx.
To evaluate the integral ∫cos^2(5x) dx, you can use a trigonometric identity. This particular integral can be simplified using the double-angle formula, which states that cos(2θ) = 1 - 2sin^2(θ).
Let's begin the evaluation:
1. Start with the original integral ∫cos^2(5x) dx.
2. Apply the double-angle identity: cos^2(θ) = (1 + cos(2θ))/2.
3. Replace θ with 5x: cos^2(5x) = (1 + cos(2*5x))/2.
4. Simplify: cos(10x) = (1 + cos(10x))/2.
5. Rewrite the integral: ∫(1 + cos(10x))/2 dx.
Now, let's compare the original integral with the answer choices to determine which one matches:
A. (Integral) 1+cos10x/2 dx
B. (Integral) 1-cos10x/2 dx
C. (Integral) 1+cos10x/20 dx
D. (Integral) 1-cos10x/20 dx
From the simplification step, we can see that the correct choice should have (1 + cos(10x))/2. Thus, the correct answer is A. (Integral) 1+cos10x/2 dx.