Which of the following is equal to
∫
−
𝜋
𝜋
𝑓
(
𝑥
)
𝑑
𝑥
correct to 3 decimal places, where
𝑓
(
𝑥
)
=
{
12
𝑥

if
−
𝜋
≤
𝑥
≤
0

3
−
3
cos
⁡
𝑥

if
0

Question

Which of the following is equal to
∫
−
𝜋
𝜋
𝑓
(
𝑥
)
𝑑
𝑥
correct to 3 decimal places, where
𝑓
(
𝑥
)
=
{
12
𝑥
	
if 
−
𝜋
≤
𝑥
≤
0

3
−
3
cos
⁡
𝑥
	
if 
0
<
𝑥
≤
𝜋
.

BlackTom AI · Accepted Answer

Integrate 12x on [-π,0] and 3 − 3 cos x on (0,π], then add the two results.

1) Relevant concepts
- Definite integrals over adjacent intervals add: ∫_{−π}^{π} f = ∫_{−π}^{0} f + ∫_{0}^{π} f.
- Basic antiderivatives: ∫ x dx = x^2/2, ∫ cos x dx = sin x.
- Careful evaluation and arithmetic (use π ≈ 3.141592653589793).

2) Step-by-step reasoning and intermediate steps
Compute the left piece:
∫_{−π}^{0} 12x dx
Antiderivative: 12 * (x^2/2) = 6x^2.
Evaluate: 6x^2 |_{x=-π}^{0} = 6(0^2) − 6((-π)^2) = 0 − 6π^2 = −6π^2.

Compute the right piece:
∫_{0}^{π} (3 − 3 cos x) dx = ∫_{0}^{π} 3 dx − 3 ∫_{0}^{π} cos x dx.
First term: 3x |_{0}^{π} = 3π − 0 = 3π.
Second term: 3 sin x |_{0}^{π} = 3(sin π − sin 0) = 3(0 − 0) = 0.
So the right piece = 3π.

Add the two:
Total integral = (−6π^2) + 3π.

3) Numerical calculation (all intermediate values)
π ≈ 3.141592653589793
π^2 ≈ 9.869604401089358
−6π^2 ≈ −6 × 9.869604401089358 = −59.21762640653615
3π ≈ 3 × 3.141592653589793 = 9.42477796076938
Sum ≈ −59.21762640653615 + 9.42477796076938 = −49.79284844576677

Round to three decimal places: −49.79284844576677 → −49.793.

4) Why this answer is correct
Each piece was integrated with the correct antiderivative and properly evaluated; the negative large contribution from the quadratic term on the left dominates the smaller positive sinusoidal-based term on the right, producing the negative total above.

5) Why other answers would be incorrect
Common errors: misreading 12x as 12/x (which is undefined at 0), forgetting the negative sign from evaluating 6x^2 at −π, or mishandling the ∫ cos x term. Any of those produce different (incorrect) numerical values.

6) Conclusion
The definite integral evaluates to −49.793 (correct to three decimal places), which matches the provided answer.

Which of the following is equal to ∫ − 𝜋 𝜋 𝑓 ( 𝑥 ) 𝑑 𝑥 correct to 3 decimal places, where 𝑓 ( 𝑥 ) = { 12 𝑥 if − 𝜋 ≤ 𝑥 ≤ 0 3 − 3 cos ⁡ 𝑥 if 0 < 𝑥 ≤ 𝜋 .单项选择题

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