Problem 34 - Calculus

Question

Find antiderinatives for the functions $$ (\sin x) e^{\cos x}+(\cos x) e^{\sin x} $$

Step-by-Step Solution

Answer

The antiderivative is \(-e^{\cos x} + e^{\sin x} + C\).

Step 1: Identify the Components of the Function

The given function is \((\sin x) e^{\cos x}+(\cos x) e^{\sin x}\). It is composed of two parts: \((\sin x) e^{\cos x}\) and \((\cos x) e^{\sin x}\). We need to find the antiderivates of both parts individually.

Step 2: Solve for the First Antiderivative

To find the antiderivative of \((\sin x) e^{\cos x}\), recognize that it involves a function of the form \(f'(x) e^{f(x)}\), which integrates to \(e^{f(x)}\). Here, \(f(x) = \cos x\) and \(f'(x) = -\sin x\). Therefore, \(\int (\sin x) e^{\cos x} \, dx = -e^{\cos x} + C_1\), where \(C_1\) is a constant.

Step 3: Solve for the Second Antiderivative

Consider \((\cos x) e^{\sin x}\) similarly to the previous step. It is of the form \(g'(x) e^{g(x)}\), which integrates to \(e^{g(x)}\). In this case, \(g(x) = \sin x\) and \(g'(x) = \cos x\). Thus, \(\int (\cos x) e^{\sin x} \, dx = e^{\sin x} + C_2\), where \(C_2\) is a constant.

Step 4: Combine the Antiderivatives

Add the results of the previous steps together to obtain the antiderivative of the entire function. Combine the constants \(C_1\) and \(C_2\) into a single constant \(C\). Therefore, the antiderivative of \((\sin x) e^{\cos x} + (\cos x) e^{\sin x}\) is \(-e^{\cos x} + e^{\sin x} + C\).

Key Concepts

Integration Techniques

Integration techniques are essential tools used to find antiderivatives, which are essentially the reverse processes of differentiation. When solving the given function \[ (\sin x) e^{\cos x} + (\cos x) e^{\sin x}, \]we incorporate specific techniques such as recognizing patterns and substitution.

Consider the technique used in solving each part of the function. We look for a pattern like \[ f'(x) e^{f(x)}, \]which allows us to directly integrate by recognizing the derivative within the product. This technique significantly simplifies the integration process.

To successfully utilize these techniques, it is crucial to:

• Recognize patterns within the integrand.
• Understand the connections between functions and their derivatives.
• Apply substitutions to simplify complex expressions.

By using these methods strategically, finding the antiderivatives becomes a more manageable task.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are periodic and play a prominent role in calculus. In integration, they often appear in products with other functions, as seen in the given problem.

Understanding their derivatives is key: - The derivative of \(\sin x\) is \(\cos x\), and - The derivative of \(\cos x\) is \(-\sin x\).

These derivatives help identify patterns used during integration.

In the expression \[ (\sin x) e^{\cos x}, \]the factor \(\sin x\) is the derivative of \(-\cos x\), allowing us to apply the integration technique discussed previously.Similarly, in \[ (\cos x) e^{\sin x}, \]\(\cos x\) is the derivative of \(\sin x\).

Mastery of these relationships helps in efficiently solving integration problems involving trigonometric functions.

Exponential Functions

Exponential functions are another cornerstone of integration, particularly due to their unique properties. An exponential function with a base \(e\) and a variable exponent is often encountered in the integration process.

What makes the integration of exponential functions straightforward is their derivative property: - If you differentiate \(e^{f(x)}\), the result is \(f'(x) e^{f(x)}\). This is a direct link between the exponential function and its derivative, making it easier to spot patterns and solve integrals.

In our problem, the functions \[ e^{\cos x} \]and \[ e^{\sin x} \]appear in products with their respective derivative partners, \(\sin x\) and \(\cos x\), respectively.

By identifying these partnerships, we can integrate the expressions effortlessly according to the recognizable pattern. The robustness of exponential functions thus simplifies integration in conjunction with trigonometric functions.