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Chapter 7: Problem 20

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

Short Answer

Expert verified
\( \int (\sin x)^{e} \cos x \, dx = \frac{(\sin x)^{e+1}}{e+1} + C \).

Step by step solution

01

Identify the Integral Structure

The integral is of the form \( \int u^n \, du\), where \( u = \sin x \) and \( n = e \). We need to find the derivative \( du \) in terms of \( dx \), which is \( du = \cos x \, dx \). This simplifies our integral.
02

Apply Substitution

Let's substitute \( u = \sin x \). Then \( du = \cos x \, dx \). Therefore, the integral becomes \( \int u^e \, du \). This is a standard power rule integral.
03

Integrate using the Power Rule

Using the power rule for integration \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \), where \( n eq -1 \), we find the integral \( \int u^e \, du \) to be \( \frac{u^{e+1}}{e+1} + C \).
04

Back-substitute \( u \)

Replace \( u \) with \( \sin x \) in the result. Thus, the integral becomes \( \frac{(\sin x)^{e+1}}{e+1} + C \).
05

Final Review and Simplify

Ensure the final expression is simplified correctly and matches the problem's requirements. The final result is \( \frac{(\sin x)^{e+1}}{e+1} + C \), where \( C \) is the constant of integration.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method in Integration
The substitution method is a powerful technique used in integration, particularly useful for simplifying integrals by making a substitution. In our problem, it helps us transform a complex integral into a simpler one. The essence of this method is to identify a part of the integral which when substituted, reduces the complexity of the expression.

Here's how it works:
  • Identify a substitution that simplifies the integrand. In our example, the substitution was letting \( u = \sin x \), which made the integral easier to handle.
  • Calculate \( du \), the differential of your substitution variable. For \( u = \sin x \), we found \( du = \cos x \, dx \).
  • Re-write the entire integral in terms of \( u \). This transformed our original integral \( \int (\sin x)^{e} \cos x \, dx \) into a simpler form \( \int u^{e} \, du \).
This method is particularly helpful when dealing with integrals involving powers, trigonometric functions, or exponential functions. It effectively reduces the problem to a more manageable form, allowing for straightforward application of integration rules.
Power Rule in Integration
The power rule for integration is a basic, yet essential tool. It applies to integrals of the form \( \int u^n \, du \), where \( n \) is a real number.

Here's the formula:
  • If \( n eq -1 \), then \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \).
  • \( C \) represents the constant of integration, crucial for accounting for all possible indefinite integrals.
In our example, after substituting \( u = \sin x \), the task was to integrate \( \int u^e \, du \). By applying the power rule, we found:
\[ \int u^e \, du = \frac{u^{e+1}}{e+1} + C \]This step is simple yet critical, ensuring we handle powers correctly when integrating functions. The power rule turns a daunting task into a straightforward arithmetic operation.
Definite and Indefinite Integrals
Integrals are categorized into two types: definite and indefinite.

Indefinite Integrals
  • These yield a family of functions, given by \( f(x) + C \), where \( C \) is the integration constant.
  • Indefinite integrals do not have set bounds; they describe general antiderivatives.
  • Our problem focuses on finding an indefinite integral, illustrated by the final result \( \frac{(\sin x)^{e+1}}{e+1} + C \).
Definite Integrals
  • These calculate the area under a curve between two points, providing a specific numerical result.
  • Definite integrals have limits and do not include a constant of integration.
  • While our exercise doesn't include one, understanding the difference is crucial for applying integration properly.
Both types of integrals are fundamental in calculus, each serving distinct purposes in analysis and real-world applications.

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Most popular questions from this chapter

Let \(R\) be a positive number. Show that $$ \int_{0}^{\pi / 2} e^{-R \sin x} d x<\frac{\pi}{2 R} $$ which is an inequality known as Jordan's inequality.

Find the particular solution of the separable differential equation that satisfies the initial condition. $$ \sqrt{x^{2}+1} \frac{d y}{d x}=\frac{x}{y} ; y(\sqrt{3})=2 $$

Find the general solution of the separable differential equation. $$ \frac{d y}{d x}=\frac{x}{y} $$

Find the particular solution of the separable differential equation that satisfies the initial condition. $$ (\ln y)^{2} \frac{d y}{d x}=x^{2} y ; y(2)=1 $$

Differentiate the function. \(f(x)=\cosh \left(\tan e^{2 x}\right)\)
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