Problem 30

Question

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

Step-by-Step Solution

Verified

Answer

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

1Step 1: Recognize the Integral Form

The integral $ \int \sin^{3} \alpha \cos \alpha \ d \alpha $ can be solved by recognizing it as a potential candidate for substitution. Notice that it's in the form of $ \sin^{n} \alpha \cos \alpha $, which suggests using a substitution method.

2Step 2: Choose a Suitable Substitution

Let $ u = \sin \alpha $. Then, $ \frac{du}{d\alpha} = \cos \alpha $ or $ du = \cos \alpha \ d\alpha $. Rewriting the integral in terms of $ u $, we get $ \int u^{3} \ du $.

3Step 3: Perform the Integration

The integral $ \int u^{3} \ du $ is straightforward. The result is $ \frac{u^{4}}{4} + C $, where $ C $ is the constant of integration.

4Step 4: Substitute Back in Terms of Original Variable

Substitute back $ u = \sin \alpha $ into the solution. This gives us $ \frac{(\sin \alpha)^{4}}{4} + C $.

5Step 5: Verify by Differentiation

Differentiate $ \frac{(\sin \alpha)^{4}}{4} + C $ with respect to $ \alpha $. Using the chain rule, the derivative is $ \sin^{3} \alpha \cos \alpha $, which matches the original integrand. Thus, the integration was performed correctly.

Key Concepts

Substitution MethodTrigonometric IntegrationVerification by Differentiation

Substitution Method

The substitution method is a strategic technique for evaluating integrals that involves changing variables to simplify the problem. In essence, it transforms a complex integral into an easier one by a clever redefinition of variables.

First, identify which part of the integrand can be substituted with a new variable. This usually becomes evident if the integral resembles a familiar form where substitution has been previously successful.
For the given problem, we see that it is in the form of $ \sin^{n} \alpha \cos \alpha $, suggesting substitution.
Choose a substitution that will simplify the integral; here, $ u = \sin \alpha $ is ideal because its derivative $ du = \cos \alpha \, d\alpha $ is already part of the integrand.
Rewrite the integral in terms of $ u $ and $ du $. This simplifies the equation to $ \int u^{3} \, du $. This integral is easier to solve than the original one.

Once substitution is complete, solve the simpler integral, and substitute back the original variable at the end of the process.

Trigonometric Integration

Trigonometric integration involves the integration of non-linear trigonometric functions. It often requires special techniques or identities for solving.

In the given exercise, the integral $ \int \sin^{3} \alpha \cos \alpha \, d\alpha $ suggests an intertwining of sine and cosine functions. Such integrals are common in problems involving trigonometric identities.

Using the substitution $ u = \sin \alpha $ means integrating the simplified polynomial $ u^{3} $ with respect to $ u $. This approach often allows us to bypass the complexities of multiple trigonometric functions and simplify it into a polynomial form for easier handling.

Recognize the integral's ability to be simplified by identifying trigonometric patterns or identities.
Use substitutions to convert these into polynomial forms.
Key to success with trigonometric integration is practice and familiarity with different types of functions and substitution tactics.

Once integrated, revert to the initial trigonometric expression to conclude the solution.

Verification by Differentiation

Verification by differentiation is a crucial step to ensure the accuracy of an integral solution. It involves differentiating the result to see if you return to the original integrand.

This approach acts as a check to confirm the integrity of your integration process:

After integration, you get a solution in terms of certain expressions like $ \frac{(\sin \alpha)^{4}}{4} + C $ in our case.
Differentiate this expression with respect to the original variable $ \alpha $.
The derivative should match the original integrand, which was $ \sin^{3} \alpha \cos \alpha $ here.
If it doesn’t match, re-evaluate the integration steps as there could be a mistake in either computation or substitution.

This step ensures that each result is not only theoretically correct, but also practically verified, guarding against errors and enhancing understanding of the integral's behavior.

Problem 29

Problem 30

Other exercises in this chapter

Problem 29

In this problem, you will show that the following improper integral converges to 1: $$ \int_{1}^{\infty} \frac{1}{x^{2}} d x $$ (a) Use the Fundamental Theorem

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Problem 29

Find an antiderivative. $$p(t)=t^{3}-\frac{t^{2}}{2}-t$$

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Problem 30

(a) Graph $f(x)=e^{-x^{2}}$ and shade the area represented by the improper integral $\int_{-\infty}^{\infty} e^{-x^{2}} d x$ (b) Use a calculator or compute

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Problem 30

Find an antiderivative. $$r(t)=\frac{1}{t^{2}}$$

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