Work out
∫
3
cos
⁡
(
2
𝜋
𝑥
)
𝑑
𝑥

Question

Work out 
∫
3
cos
⁡
(
2
𝜋
𝑥
)
𝑑
𝑥

BlackTom AI · Accepted Answer

The antiderivative of cos(kx) is (1/k) sin(kx), so we will use that property with k = 2π.

1. Relevant knowledge points and concepts
- Linearity of the integral: constants can be pulled outside the integral.
- Basic antiderivative: ∫ cos(u) du = sin(u) + C.
- Chain rule (in reverse): ∫ cos(kx) dx = (1/k) sin(kx) + C because d/dx[sin(kx)] = k cos(kx).
- Substitution method: let u = kx to justify the 1/k factor.

2. Complete reasoning process step by step
Start with the integrand 3 cos(2πx). Use linearity to factor out 3:
∫ 3 cos(2πx) dx = 3 ∫ cos(2πx) dx.
Evaluate ∫ cos(2πx) dx by substitution: let u = 2πx, so du = 2π dx, hence dx = du/(2π).

3. Calculational intermediate steps
3 ∫ cos(2πx) dx
= 3 ∫ cos(u) * (du/(2π))
= 3/(2π) ∫ cos(u) du
= 3/(2π) sin(u) + C
Replace u = 2πx:
= (3/(2π)) sin(2πx) + C.

4. Why the correct answer is correct
Differentiate the result to verify:
d/dx [(3/(2π)) sin(2πx)] = (3/(2π)) * cos(2πx) * 2π = 3 cos(2πx),
which reproduces the original integrand. Thus the antiderivative is correct.

5. Why other possible answers are incorrect
- Omitting the 1/(2π) factor (i.e., writing 3 sin(2πx) + C) is wrong because differentiation would introduce the extra factor 2π and not match the integrand.
- Introducing a negative sign (e.g., -3/(2π) sin(2πx)) is incorrect because d/dx[sin] = cos, not -cos.
- Any constant multiple different from 3/(2π) fails verification by differentiation.

6. Conclusion
The integral evaluates to (3/(2π)) sin(2πx) + C, which matches the provided answer.

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