| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To find the indefinite integral of the function \(-\sin(x+5)\), we will use the substitution method. Let's proceed step by step. 1. **Substitution**: Let \( u = x + 5 \). Then, \( du = dx \). 2. **Rewrite the integral**: Substitute \( u \) and \( du \) into the integral: \[ \int -\sin(x+5) \, dx = \int -\sin(u) \, du \] 3. **Integrate**: The integral of \(-\sin(u)\) with respect to \( u \) is: \[ \int -\sin(u) \, du = \cos(u) + C \] 4. **Substitute back**: Replace \( u \) with \( x + 5 \): \[ \cos(u) + C = \cos(x+5) + C \] Thus, the indefinite integral of \(-\sin(x+5)\) is: \[ \boxed{\cos(x+5) + C} \] |
\cos (5) \cos (x)-\sin (5) \sin (x) |
| deepseek-r1 | To find the indefinite integral of \(-\sin(x + 5)\), we start by recognizing that the integral of \(\sin(u)\) is \(-\cos(u) + C\). Here, we can use substitution to handle the argument \(x + 5\). 1. **Substitution**: Let \(u = x + 5\). Then, \(du = dx\). 2. **Rewrite the integral**: The integral becomes \(-\int \sin(u) \, du\). 3. **Integrate**: The integral of \(\sin(u)\) is \(-\cos(u)\), so we have: \[ -\int \sin(u) \, du = -(-\cos(u)) + C = \cos(u) + C \] 4. **Substitute back**: Replace \(u\) with \(x + 5\) to get \(\cos(x + 5) + C\). To verify, we differentiate \(\cos(x + 5)\) and check that we obtain the original integrand: - The derivative of \(\cos(x + 5)\) is \(-\sin(x + 5)\), which matches the original function \(-\sin(x + 5)\). Thus, the indefinite integral of \(-\sin(x + 5)\) is \(\boxed{\cos(x + 5)} + C\). |
\cos (5) \cos (x)-\sin (5) \sin (x) |
| gpto1 | To find the indefinite integral of the function \(-\sin(x+5)\), we can proceed as follows: First, factor out the negative sign: \[ \int -\sin(x+5)\,dx = -\int \sin(x+5)\,dx \] Next, use a substitution to simplify the integral. Let: \[ u = x + 5 \\ du = dx \] Now the integral becomes: \[ -\int \sin(u)\,du \] Recall that the integral of \(\sin(u)\) with respect to \(u\) is \(-\cos(u)\): \[ -\int \sin(u)\,du = -(-\cos(u)) = \cos(u) \] Substitute back \(u = x + 5\): \[ \cos(u) = \cos(x + 5) \] So, the indefinite integral is: \[ \int -\sin(x+5)\,dx = \cos(x+5) + C \] Since any constant \(C\) can be added (the integral can vary by a constant), the final answer is: \[ \boxed{\cos(x+5)} \] \boxed{\cos(x+5)} |
\cos (5) \cos (x)-\sin (5) \sin (x) |
| gpt-4o-2024-08-06 | To find the indefinite integral of the function \(-\sin(x+5)\), we need to determine the antiderivative. The integral we need to solve is: \[ \int -\sin(x+5) \, dx \] We know that the antiderivative of \(\sin(u)\) is \(-\cos(u)\). Therefore, the antiderivative of \(-\sin(u)\) is \(\cos(u)\). In this case, let \(u = x + 5\). Then, the derivative \(du = dx\), so we can directly integrate: \[ \int -\sin(x+5) \, dx = \cos(x+5) + C \] where \(C\) is the constant of integration. Thus, the indefinite integral of \(-\sin(x+5)\) is: \[ \boxed{\cos(x+5) + C} \] |
\cos (5) \cos (x)-\sin (5) \sin (x) |
| DeepSeek-Coder-V2-Lite-Instruct | To find the indefinite integral of the function \(-\sin(x+5)\), we will use the method of substitution. Let's proceed step-by-step: 1. **Substitution**: Let \( u = x + 5 \). Then, \( du = dx \). 2. **Rewrite the integral**: Substitute \( u \) and \( du \) into the integral: \[ \int -\sin(x+5) \, dx = \int -\sin(u) \, du \] 3. **Integrate**: The integral of \(-\sin(u)\) with respect to \(u\) is: \[ \int -\sin(u) \, du = \cos(u) + C \] where \( C \) is the constant of integration. 4. **Substitute back**: Replace \( u \) with \( x + 5 \): \[ \cos(u) + C = \cos(x+5) + C \] Thus, the indefinite integral of \(-\sin(x+5)\) is: \[ \boxed{\cos(x+5) + C} \] |
\cos (5) \cos (x)-\sin (5) \sin (x) |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: Recall the basic integral of the sine function The integral of $\sin(x)$ is $-\cos(x) + C$, where $C$ is the constant of integration. ## Step 2: Apply the substitution method for the given function Given the function $-\sin(x+5)$, we can use the substitution $u = x + 5$ to simplify the integral. ## Step 3: Find the derivative of u The derivative of $u$ with respect to $x$ is $\frac{du}{dx} = 1$. ## Step 4: Rewrite the integral in terms of u The integral becomes $-\int \sin(u) \frac{du}{dx} dx = -\int \sin(u) du$. ## Step 5: Integrate the function with respect to u Using the basic integral of the sine function, we get $-\int \sin(u) du = \cos(u) + C$. ## Step 6: Substitute back u = x + 5 Substituting back $u = x + 5$, we get $\cos(x + 5) + C$. ## Step 7: Write the final answer in the required format The final answer is: $\boxed{\cos(x + 5) + C}$ |
\cos (5) \cos (x)-\sin (5) \sin (x) |
| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Recall the general form of the integral of the sine function The integral of $\sin(u)$ with respect to $u$ is $-\cos(u) + C$, where $C$ is the constant of integration. ## Step 2: Identify the argument of the sine function in the given problem The argument of the sine function is $x+5$. ## Step 3: Apply the general form of the integral of the sine function to the given problem Substitute $u = x+5$ into the general form of the integral of the sine function. The integral of $-\sin(x+5)$ with respect to $x$ is $-\cos(x+5) + C$, where $C$ is the constant of integration. ## Step 4: Write the final answer in the required format The final answer is: $\boxed{-\cos(x+5) + C}$ |
\cos (5) \cos (x)-\sin (5) \sin (x) |
| Qwen2-72B-Instruct | To find the indefinite integral of the function $-\sin(x + 5)$, we use the basic rules of integration. The integral of $\sin(x)$ is $-\cos(x)$, and since we have a linear argument inside the sine function, we apply the chain rule for integration. The integral of $-\sin(x + 5)$ is: \[ \int -\sin(x + 5) \, dx = \cos(x + 5) + C \] where $C$ is the constant of integration. So, the final answer is: \[ \boxed{\cos(x + 5) + C} \] |
\cos (5) \cos (x)-\sin (5) \sin (x) |
| baichuan4 | To find the indefinite integral of the function $-\sin(x+5)$, we can use the fact that the integral of a sum is the sum of the integrals, and the integral of a constant times a function is the constant times the integral of the function. We also know that the integral of the sine function is the negative cosine function, and the derivative of the cosine function is the negative sine function. This means that the integral of $-\sin(x)$ is $\cos(x)$. Applying this knowledge to our function, we can see that: $$ \int -\sin(x+5) \, dx = -\int \sin(x+5) \, dx $$ To integrate $\sin(x+5)$, we can use the substitution rule. Let $u = x + 5$, then $du = dx$. The integral becomes: $$ -\int \sin(u) \, du $$ Now, we can integrate $\sin(u)$ directly: $$ -\int \sin(u) \, du = -(-\cos(u)) = \cos(u) $$ Finally, we substitute back $u = x + 5$: $$ \cos(u) = \cos(x + 5) $$ And since the integral was indefinite, we add the constant of integration $C$: $$ \int -\sin(x+5) \, dx = \cos(x + 5) + C $$ Putting the final answer in a box, as requested: $$ \boxed{\cos(x + 5) + C} $$ This is the indefinite integral of the function $-\sin(x+5)$. |
\cos (5) \cos (x)-\sin (5) \sin (x) |
| doubao | The integral of $-\sin(x+5)$ is $\cos(x+5) + C$. So the answer is $\boxed{\cos(x+5) + C}$. |
\cos (5) \cos (x)-\sin (5) \sin (x) |
| gpto1 | To find the indefinite integral of the function \(-\sin(x+5)\), we can proceed as follows: First, factor out the negative sign: \[ \int -\sin(x+5)\,dx = -\int \sin(x+5)\,dx \] Next, use a substitution to simplify the integral. Let: \[ u = x + 5 \\ du = dx \] Now the integral becomes: \[ -\int \sin(u)\,du \] Recall that the integral of \(\sin(u)\) with respect to \(u\) is \(-\cos(u)\): \[ -\int \sin(u)\,du = -(-\cos(u)) = \cos(u) \] Substitute back \(u = x + 5\): \[ \cos(u) = \cos(x + 5) \] So, the indefinite integral is: \[ \int -\sin(x+5)\,dx = \cos(x+5) + C \] Since any constant \(C\) can be added (the integral can vary by a constant), the final answer is: \[ \boxed{\cos(x+5)} \] \boxed{\cos(x+5)} |
\cos (5) \cos (x)-\sin (5) \sin (x) |
| kimi | To find the indefinite integral of the function \(-\sin(x+5)\), we will use the substitution method. Let's proceed step by step. 1. **Substitution:** Let \( u = x + 5 \). Then, \( du = dx \). 2. **Rewrite the integral:** Substitute \( u \) and \( du \) into the integral: \[ \int -\sin(x+5) \, dx = \int -\sin(u) \, du \] 3. **Integrate:** The integral of \(-\sin(u)\) with respect to \( u \) is: \[ \int -\sin(u) \, du = -\cos(u) + C \] where \( C \) is the constant of integration. 4. **Back-substitute:** Now, substitute back \( u = x + 5 \) into the result: \[ -\cos(u) + C = -\cos(x+5) + C \] 5. **Final answer:** The indefinite integral of \(-\sin(x+5)\) is: \[ \boxed{-\cos(x+5) + C} \] |
\cos (5) \cos (x)-\sin (5) \sin (x) |
| minimax | To find the indefinite integral of the function $-\sin(x+5)$, we can use the basic integration rule for sine functions, which states that the integral of $\sin(ax)$ with respect to $x$ is $-\frac{1}{a}\cos(ax) + C$, where $C$ is the constant of integration. Applying this rule to the given function, we treat $(x+5)$ as our $ax$ term, where $a = 1$. Thus, we have: $$ \int -\sin(x+5) \, dx = -\int \sin(x+5) \, dx = -\left(-\frac{1}{1}\cos(x+5)\right) + C $$ Simplifying this, we get: $$ \int -\sin(x+5) \, dx = \cos(x+5) + C $$ So the final answer, in the boxed format, is: $$ \boxed{\cos(x+5) + C} $$ |
\cos (5) \cos (x)-\sin (5) \sin (x) |
| wenxinyiyan | 首先，我们需要找到$-\sin(x+5)$的一个原函数。根据微积分基本定理，如果$F(x)$是$f(x)$的一个原函数，那么$\int f(x) \, dx = F(x) + C$，其中$C$是积分常数。 对于$-\sin(x+5)$，其一个原函数可以通过反导数求得。我们知道$\frac{d}{dx}\cos(x+5) = -\sin(x+5)$，因此$-\sin(x+5)$的一个原函数是$\cos(x+5)$。 所以，$-\sin(x+5)$的不定积分为： $$ \int -\sin(x+5) \, dx = \cos(x+5) + C $$ |
\cos (5) \cos (x)-\sin (5) \sin (x) |
| zhipu | To find the indefinite integral of the function \(-\sin(x+5)\), we can use the basic integration rule for the sine function. The integral of \(\sin(u)\) with respect to \(u\) is \(-\cos(u)\). Here, \(u = x + 5\). Let's proceed with the integration: \[ \int -\sin(x+5) \, dx \] We can factor out the constant \(-1\): \[ = - \int \sin(x+5) \, dx \] Now, we perform a substitution. Let \(u = x + 5\). Then, \(du = dx\). The integral becomes: \[ = - \int \sin(u) \, du \] Using the integral rule for \(\sin(u)\): \[ = - (-\cos(u)) + C \] Simplifying the expression: \[ = \cos(u) + C \] Substitute back \(u = x + 5\): \[ = \cos(x + 5) + C \] Therefore, the indefinite integral of \(-\sin(x+5)\) is: \[ \boxed{\cos(x+5) + C} \] |
\cos (5) \cos (x)-\sin (5) \sin (x) |
| deepseek-v3 | To find the indefinite integral of the function \( -\sin(x + 5) \), we can follow these steps: 1. **Identify the Integral:** \[ \int -\sin(x + 5) \, dx \] 2. **Factor Out the Constant:** The negative sign is a constant and can be factored out of the integral: \[ -\int \sin(x + 5) \, dx \] 3. **Integrate the Sine Function:** The integral of \( \sin(u) \) with respect to \( u \) is \( -\cos(u) \). Here, \( u = x + 5 \), so: \[ \int \sin(x + 5) \, dx = -\cos(x + 5) + C \] where \( C \) is the constant of integration. 4. **Apply the Factored Constant:** Multiply the result by the negative sign that was factored out earlier: \[ -\left(-\cos(x + 5) + C\right) = \cos(x + 5) - C \] 5. **Simplify the Constant:** Since \( C \) is an arbitrary constant, \( -C \) can be replaced with a new constant \( C' \): \[ \cos(x + 5) + C' \] 6. **Final Answer:** \[ \boxed{\cos(x + 5) + C} \] Here, \( C \) represents the constant of integration, which can vary. |
\cos (5) \cos (x)-\sin (5) \sin (x) |