Problem 26 Find the integrals in problems. ... [FREE SOLUTION]

91影视

Applied Calculus

Deborah Hughes-Hallett, Patti Frazer Lock, Andrew M. Gleason

$Math Studyset 91影视 Explanations$ Math

4 Edition

Chapter 7: Problem 26

Find the integrals in problems. Check your answers by differentiation. $$ \int \sin ^{3} \alpha \cos \alpha d \alpha $$

Short Answer

Expert verified

The integral is $ \frac{(\sin \alpha)^4}{4} + C $.

Step by step solution

Identify the Integrand Type and Substitution

Upon looking at the integrand $ \sin^3 \alpha \cos \alpha $, we notice that it can be simplified using a substitution method. Let $ u = \sin \alpha $. Then, $ du = \cos \alpha \, d\alpha $. This substitution transforms the integral into $ \int u^3 \, du $.

Perform the Integration

Now, integrate $ \int u^3 \, du $ using the power rule for integration. The power rule states that $ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C $. Here $ n = 3 $, so the integral becomes:\[ \int u^3 \, du = \frac{u^4}{4} + C \].

Reverse Substitution

Substitute back $ u = \sin \alpha $ into the integrated function. We have:\[ \frac{(\sin \alpha)^4}{4} + C \].

Differentiate to Check

To confirm our integration, we differentiate $ \frac{(\sin \alpha)^4}{4} + C $ with respect to $ \alpha $.Using the chain rule, let $ v = \sin \alpha $, then $ \frac{dv}{d\alpha} = \cos \alpha $, and$ \frac{d}{d\alpha} \left( \frac{v^4}{4} \right) = v^3 \cdot \frac{dv}{d\alpha} = \sin^3 \alpha \cdot \cos \alpha $,which matches the original integrand.

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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 technique used to simplify integrals, making them easier to solve. The idea is to replace a more complicated expression with a simpler one.

Let's take our example integrand, $ \sin^3 \alpha \cos \alpha $, as a case study. Here, the expression is difficult to integrate directly, so we use the substitution method. We set $ u = \sin \alpha $ and find $ du = \cos \alpha \, d\alpha $.

By substituting $ u $ for $ \sin \alpha $ and $ du $ for $ \cos \alpha \, d\alpha $, the integral $ \int \sin^3 \alpha \cos \alpha \, d\alpha $ simplifies to $ \int u^3 \, du $.
This method transforms a complex trigonometric integral into a simple polynomial one.

So, the substitution method is all about making the problem easier to manage by "substituting" complicated parts with simpler variables.

Power Rule

The power rule is a fundamental rule in calculus used to integrate functions of the form $ x^n $. Understanding the power rule is essential for integrating polynomial expressions.

For any function $ \int x^n \, dx $, the power rule tells us that the integral is given by:\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C,\]where $ C $ is the constant of integration. It's important to note that $ n eq -1 $ because the power rule does not apply to the integral of $ x^{-1} $.

In our worked example, once we applied the substitution $ u = \sin \alpha $, our integral became $ \int u^3 \, du $.
Using the power rule, this simplifies to $ \frac{u^4}{4} + C $.

The power rule is thus a straightforward technique you can use to find the integral of any polynomial function of the form $ x^n $.

Chain Rule

While the chain rule is primarily a differentiation technique, it assists in confirming our integration results by checking our work. The chain rule states that if you have a function $ y $ that depends on $ u $, and $ u $ itself is a function of $ x $, then:\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}. \]

This rule is applied during the "check" process in integration. Let's see how it was used in our example:

After finding the integral $ \frac{(\sin \alpha)^4}{4} + C $, we used the chain rule to differentiate it with respect to $ \alpha $.
By letting $ v = \sin \alpha $, we differentiate $ \frac{v^4}{4} $ through $ \frac{dy}{dv} = v^3 $ and multiplying by $ \frac{dv}{d\alpha} = \cos \alpha $.

By doing so, we end up with the original integrand $ \sin^3 \alpha \cos \alpha $, confirming that our integration was done correctly.
The chain rule thus serves as a valuable tool for ensuring the accuracy of our integration by differentiation.

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91影视

Find the integrals in problems. Check your answers by differentiation. $$ \int \sin ^{3} \alpha \cos \alpha d \alpha $$

Short Answer

Step by step solution

Identify the Integrand Type and Substitution

Perform the Integration

Reverse Substitution

Differentiate to Check

Key Concepts

Substitution Method

Power Rule

Chain Rule

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