ChatGPT에게 적분 문제 풀이를 요청해봅시다.
[!user] User
$4\displaystyle \int_0 ^{\frac\pi3} (8\cos^3 \theta - 1)\cos\theta d\theta$.
[!gpt] ChatGPT The integral of $4 \int_0^{\pi/3} (8\cos^3 \theta - 1)\cos\theta \, d\theta$ evaluates to $\frac{3 \sqrt{3}}{2} + 4\pi$.
계산 결과를 알려주었지만 식 전개 과정을 설명해주지 않았으므로, 이를 요청해봅시다.
[!user] User explain step by step
[!gpt] ChatGPT Certainly! Let’s break down the integral step by step:
The integral we are evaluating is:
\[4 \int_0^{\pi/3} (8\cos^3(\theta) - 1)\cos(\theta) \, d\theta\]
Expand the integrand: First, we expand the integrand by multiplying $\cos(\theta)$ with each term inside the parentheses.
\[4 \int_0^{\pi/3} (8\cos^3(\theta)\cos(\theta) - \cos(\theta)) \, d\theta\]This simplifies to:
\[4 \int_0^{\pi/3} (8\cos^4(\theta) - \cos(\theta)) \, d\theta\]Split the integral: We can split this integral into two separate integrals:
\[4 \left[ \int_0^{\pi/3} 8\cos^4(\theta) \, d\theta - \int_0^{\pi/3} \cos(\theta) \, d\theta \right]\]Evaluate each integral:
For the first integral, $\int_0^{\pi/3} 8\cos^4(\theta) \, d\theta$, we would typically use a power-reducing formula to simplify $\cos^4(\theta)$ into terms involving $\cos(2\theta)$ or $\cos(4\theta)$, and then integrate.
For the second integral, $\int_0^{\pi/3} \cos(\theta) \, d\theta$, the antiderivative of $\cos(\theta)$ is $\sin(\theta)$, so we evaluate $\sin(\theta)$ from $0$ to $\pi/3$.
Combine the results: After evaluating both integrals, we combine the results and multiply by 4 (as per the original integral).
Final calculation: The final step is to perform the actual calculations, which involves substituting the limits of integration and simplifying the expression.
The result of these steps gives us the final answer, which is $\frac{3 \sqrt{3}}{2} + 4\pi$. This process involves some advanced calculus techniques, particularly for the integral involving $\cos^4(\theta)$, which is why the direct computation is often preferred.
ChatGPT가 자세한 설명을 해주었지만, 수식보다는 문장으로 상황을 묘사하고 있어 가독성이 떨어집니다. 각 단계의 수식을 달라고 요청해봅시다.
[!user] User give me each step with mathematical expression
[!gpt] ChatGPT Certainly! Let’s go through each step with the corresponding expressions:
Expand the integrand: Integral: $4 \int_0^{\pi/3} (8\cos^3(\theta) - 1)\cos(\theta) \, d\theta$ Expanded: $4 \int_0^{\pi/3} (8\cos^4(\theta) - \cos(\theta)) \, d\theta$
Split the integral: Split into two integrals: $4 \left[ \int_0^{\pi/3} 8\cos^4(\theta) \, d\theta - \int_0^{\pi/3} \cos(\theta) \, d\theta \right]$
- Evaluate each integral:
For $\int_0^{\pi/3} 8\cos^4(\theta) \, d\theta$: We use the power-reducing formula for $\cos^4(\theta)$. The power-reducing formula is:
\[\cos^4(\theta) = \frac{1}{8} \left[ 3 + 4\cos(2\theta) + \cos(4\theta) \right]\]So, the integral becomes:
\[\int_0^{\pi/3} 8 \left[ \frac{1}{8} \left( 3 + 4\cos(2\theta) + \cos(4\theta) \right) \right] \, d\theta\]Simplifying and integrating term by term gives us the result for this part.
For $\int_0^{\pi/3} \cos(\theta) \, d\theta$:
The antiderivative of $\cos(\theta)$ is $\sin(\theta)$, so we evaluate: \(\sin(\theta) \Big|_0^{\pi/3}\) This gives $\sin(\pi/3) - \sin(0)$.
Combine the results:
After evaluating both integrals, we add them together and multiply by 4.
Final calculation:
Perform the actual calculations, including substituting the limits of integration and simplifying the expression.
The detailed calculations, especially for the first integral involving $\cos^4(\theta)$, require a bit of work. The power-reducing step is particularly important as it simplifies the integral into a form that can be easily integrated.
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