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Chapter 4: Problem 48

Find the indefinite integral. $$ \int e^{\sin x} \cos x d x $$

Short Answer

Expert verified
The short answer for the indefinite integral of \(e^{\sin x} \cos x \, dx\) is: \[\int e^{\sin x} \cos x \, dx = e^{\sin x} + C.\]

Step by step solution

01

Identifying substitution

Observe the integral expression and notice that the derivative of \(\sin x\) is \(\cos x\), which is present in the integral. This suggests that the substitution method can be used to simplify the integral. We will let \[u = \sin x\] and so \[\frac{du}{dx} = \cos x.\]
02

Substitute in the integral

Now, we have the substitution in place. We just need to change all instances of \(x\) in our integral to \(u\). Using the substitutions \(u = \sin x\) and \(du = \cos x \, dx\), we arrive at a simpler integral expression: \[\int e^u du.\]
03

Finding the antiderivative

The integral expression has been significantly simplified. Now we just need to find the antiderivative of \(e^u\). Since the derivative of \(e^u\) is simply \(e^u\) itself, the antiderivative of \(e^u\) is also \(e^u\). So, we have: \[\int e^u du = e^u + C,\] where \(C\) is the constant of integration.
04

Substitute back_x

Now that we have found the antiderivative, we need to substitute back in our original variable, \(x\). Recall that \(u = \sin x\), so we can rewrite the antiderivative expression in terms of \(x\): \[e^u + C = e^{\sin x} + C.\]
05

Final Answer

Our final answer for the indefinite integral of \(e^{\sin x} \cos x \, dx\) is: \[\int e^{\sin x} \cos x \, dx = e^{\sin x} + C.\]

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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 used in calculus to simplify integration. It involves identifying a part of the integrand (the function being integrated) that can be replaced by a new variable. This method is particularly useful when a derivative of a function is also present in the integrand, as in our example:
  • Start by identifying a substitution. In our integral, we observed that \(\sin x\)'s derivative, \(\cos x\), is present. This hints that \(u = \sin x\) is a good substitution to simplify the problem.
  • Change variables in the integral. Replace \(\sin x\) with \(u\) and \(\cos x \, dx\) with \(du\), transforming the integral to \(\int e^u \, du\).
By converting the integrand to a simpler form, integration becomes more manageable, turning a potentially complex integral into one that is straightforward.
Antiderivative
When calculating an indefinite integral, finding the antiderivative is the main goal. The antiderivative of a function is another function whose derivative is the original function. In this exercise, we needed:
  • To find the antiderivative of \(e^u\), which is already a familiar form since \(e^u\)'s derivative is itself.
  • The solution involves recognizing that the antiderivative of \(e^u\) is \(e^u + C\), where \(C\) is the constant of integration.
The antiderivative tells us the family of functions that can result in the original integrand when differentiated. Once we have it in terms of the substitution variable, we backtrack to the initial variable.
Constant of Integration
In indefinite integrals, the constant of integration, denoted as \(C\), plays a vital role. Since differentiation eliminates constants:
  • The antiderivative is not unique, unlike derivatives. Including \(C\) accounts for any number that might have been lost in differentiation.
  • This constant indicates that there are infinitely many antiderivatives, each differing by a constant.
Understanding the constant of integration ensures that when presenting the solution, you acknowledge all possible original functions.
Trigonometric Functions
In calculus, trigonometric functions come up often, extending their utility beyond basic geometry applications. They are crucial when dealing with integrals like \(\int e^{\sin x} \cos x \, dx\):
  • The problem involved \(\sin x\) and \(\cos x\), showcasing the fundamental relationships between these functions.
  • Recognizing derivatives and integrals related to trigonometric functions is essential. For example, knowing \(\frac{d}{dx}(\sin x) = \cos x\) allows strategic substitution.
A strong grasp of trigonometric identities and derivatives simplifies complex integrals, making trigonometry an integral part of calculus problem-solving.

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