Chapter 2: Problem 2
Evaluate \(\int \sin ^{4} x \cos x d x\)
Short Answer
Expert verified
\( \int \sin^4 x \cos x \, dx = \frac{\sin^5 x}{5} + C \).
Step by step solution
01
Identify the Integral Expression
We need to evaluate the integral \( \int \sin^4 x \cos x \, dx \). Notice that this integral is a standard form where we have a power of sine multiplied by a power of cosine.
02
Use Substitution Method
When we have powers of sine and cosine, it can often be useful to use a substitution involving one of the trigonometric functions. In this case, set \( u = \sin x \), which implies \( du = \cos x \, dx \). The integral then becomes \( \int u^4 du \).
03
Integrate with respect to u
Now, integrate the expression \( \int u^4 \, du \). The result of this integration is \( \frac{u^5}{5} + C \), where \( C \) is the constant of integration.
04
Substitute Back the Original Variable
Substitute back the original variable \( x \) by replacing \( u = \sin x \). This gives us \( \frac{\sin^5 x}{5} + C \).
05
Write the Final Answer
Thus, the evaluated integral is \( \frac{\sin^5 x}{5} + C \), 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.
Trigonometric Substitution
Trigonometric substitution is a clever technique used to simplify integrals involving trigonometric functions. It works by replacing a complicated part of the integral with a simpler expression. This is especially useful when dealing with expressions that contain square roots or trigonometric identities.
In our exercise, we encountered an integral involving powers of sine and cosine, \( \int \sin^4 x \cos x \, dx \). Recognizing that the power of sine can be rewritten by using a substitution is key. By setting \( u = \sin x \), we transform the integral into a polynomial in terms of \( u \), which is often easier to solve.
This substitution not only simplifies the integral but also seamlessly integrates trigonometric functions, turning them into algebraic ones. Hence, trigonometric substitution is a powerful tool when tackling integrals from trigonometric functions in calculus.
In our exercise, we encountered an integral involving powers of sine and cosine, \( \int \sin^4 x \cos x \, dx \). Recognizing that the power of sine can be rewritten by using a substitution is key. By setting \( u = \sin x \), we transform the integral into a polynomial in terms of \( u \), which is often easier to solve.
This substitution not only simplifies the integral but also seamlessly integrates trigonometric functions, turning them into algebraic ones. Hence, trigonometric substitution is a powerful tool when tackling integrals from trigonometric functions in calculus.
Trigonometric Integrals
Trigonometric integrals are those integrals that involve trigonometric functions like sine, cosine, tangent, and others. These integrals can often look intimidating at first glance due to the oscillatory nature of trig functions.
However, they can be made manageable by identifying patterns and using various techniques. For example, integrals of the form \( \int \sin^m x \cos^n x \, dx \) often allow for simplifications using trigonometric identities or strategic substitutions.
In the given exercise \( \int \sin^4 x \cos x \, dx \), we notice a specific pattern of powers of sine and cosine, inviting us to explore either substitution or trigonometric identities to simplify our work. Mastering trigonometric integrals can enhance your calculus skills, making these calculations much easier to handle.
However, they can be made manageable by identifying patterns and using various techniques. For example, integrals of the form \( \int \sin^m x \cos^n x \, dx \) often allow for simplifications using trigonometric identities or strategic substitutions.
In the given exercise \( \int \sin^4 x \cos x \, dx \), we notice a specific pattern of powers of sine and cosine, inviting us to explore either substitution or trigonometric identities to simplify our work. Mastering trigonometric integrals can enhance your calculus skills, making these calculations much easier to handle.
Substitution Method
The substitution method is a fundamental technique in integral calculus, often used to transform complex integrals into simpler forms. By swapping out one part of an integral with a new variable, we can often change the entire format of the calculation.
In this problem, the substitution method was ideal. We set \( u = \sin x \), transforming the integral to \( \int u^4 \, du \). This is easier to understand because integrating polynomials is straightforward compared to handling trigonometric expressions directly.
A key part of the substitution method is choosing a substitution that simplifies the integration process. It is followed by integrating with respect to the new variable and then substituting back to the original variable. This method can significantly simplify complex integrals, saving time and effort.
In this problem, the substitution method was ideal. We set \( u = \sin x \), transforming the integral to \( \int u^4 \, du \). This is easier to understand because integrating polynomials is straightforward compared to handling trigonometric expressions directly.
A key part of the substitution method is choosing a substitution that simplifies the integration process. It is followed by integrating with respect to the new variable and then substituting back to the original variable. This method can significantly simplify complex integrals, saving time and effort.
