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Magazine Chapter 5: Problem 31
Evaluate the integrals using appropriate substitutions. $$ \int e^{\sin x} \cos x d x $$
Short Answer
Expert verified
The evaluated integral is \( e^{\sin x} + C \).
Step by step solution
01
Identify the Substitution
To solve the integral \( \int e^{\sin x} \cos x \, dx \), observe that the exponent \( \sin x \) and its derivative \( \cos x \) are both present in the integrand. This suggests using substitution where \( u = \sin x \).
02
Differentiate the Substitution
Differentiate the substitution \( u = \sin x \) with respect to \( x \) to find \( du \):\[\frac{du}{dx} = \cos x\]Thus, \( du = \cos x \, dx \).
03
Substitute in the Integral
Substitute \( u = \sin x \) and \( du = \cos x \, dx \) into the integral:\[\int e^{\sin x} \cos x \, dx = \int e^u \, du\]
04
Integrate
Integrate \( \int e^u \, du \):\[\int e^u \, du = e^u + C\]where \( C \) is the constant of integration.
05
Substitute Back
Replace \( u \, (\sin x) \) back into the integrated result:\[e^u + C = e^{\sin x} + C\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a powerful technique for evaluating integrals, especially when dealing with composite functions. It is particularly useful when an integral contains a function and its derivative. The fundamental idea is to simplify the integral by replacing parts of it with a single variable, namely \( u \).
- First, identify a substitution that will simplify the integrand. Often, this involves choosing \( u \) to be a function inside the integrand whose derivative is also present.
- In the given exercise, we observe that the integrand involves \( e^{\sin x} \) and \( \cos x \). Noticing that the derivative of \( \sin x \) is \( \cos x \), we choose \( u = \sin x \).
- Next, differentiate \( u \) with respect to \( x \) to find \( du \). In this example, \( \frac{du}{dx} = \cos x \), thus \( du = \cos x \, dx \).
- Finally, substitute \( u \) and \( du \) back into the integral, which transforms it into an easier integral of the form \( \int e^u \, du \).
Exponential Functions
Exponential functions are central to calculus due to their unique properties, especially when involved in integration. These functions, typically expressed as \( e^x \), where \( e \) is the base of natural logarithms, have remarkable features.
- One significant property is that the derivative and integral of \( e^x \) are both \( e^x \). This self-replicating nature simplifies integration immensely.
- In the exercise, we see the function \( e^{\sin x} \), an exponential function with an inner function, which calls for substitution methods to simplify the evaluation.
- Understanding exponential functions' behavior is crucial to tackling integrals involving expressions like \( e^{something} \). Always check if a substitution can reduce the exponential to a simpler form.
- Once substituted, evaluating the integral \( \int e^u \, du \) becomes straightforward, as it integrates to \( e^u + C \).
Integration Techniques
Integration techniques serve as a toolbox for handling various integrals, whether they involve polynomials, trigonometric, or exponential functions. Among these methods, the substitution method is just one avenue to explore.
- It's vital to choose the correct technique based on the nature of the integrand. Techniques can overlap, so knowing when to apply each is key.
- Common techniques include substitution, integration by parts, partial fraction decomposition, and trigonometric identities.
- Substitution is most effective when the integrand contains a function and its derivative, as seen in \( \int e^{\sin x} \cos x \, dx \). The derivative of an inside function simplifies the process.
- For more complex integrals, sometimes combining techniques, like substitution followed by recognition of a standard integral form, is necessary.
