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Lab 4 Trig Integrals Part 1 Powers of Sine and Cosine The purpose of part one of this activity is to develop a strategy for integrating powers of Sine and Cosine, that is, integrals of the form $\int sin^n \theta cos^m \theta d\theta$. Work with your neighbor to make sense of the following problems as you work toward developing your own general strategy. Would your strategy work for any of these in the table below? Indicate which by writing the simplified form of the integrand after substitution. Problem Substitution New integral in w $\int sin^3 \theta cos \theta d\theta$ $w = sin \theta$ $\int w^3 dw$ $\int sin^5 \theta cos \theta d\theta$ $\int sin \theta cos^3 \theta d\theta$ $\int cos^4 \theta d\theta$ Describe the general form of the integrands in the table above. Briefly describe your strategy for integrating the functions in the table above 1. Evaluate $\int sin^3 \theta cos \theta d\theta$. 2. Now consider $\int sin \theta cos^3 \theta d\theta$. How is this problem different from the problems on the last page? Slick Maneuver of the Day: "Pull off" one of the powers of $cos \theta$ $cos^3 \theta = cos^2 \theta cos \theta$ and use the identity $sin^2 \theta + cos^2 \theta = 1$ to transform the problem into one of the type in problem 1. Try it yourself Would the "slick maneuver of the day" also work for the following integrals? If so, indicate the algebra maneuver, the trig identity, and the new integral after the substitution. Problem Perform Algebra Maneuver Use Trig Identity Make Substitution $\int sin^3 \theta cos \theta d\theta$ $\int sin^3 \theta cos^2 \theta d\theta$ $\int sin^3 \theta (1 - sin^2 \theta) cos \theta d\theta$ $\int w^3 (1 - w^2) dw$ $\int cos^3 \theta d\theta$ $\int sin^5 \theta d\theta$ $\int sin^3 \theta cos^2 \theta d\theta$ Wait a minute, would the strategy outlined above work for the following problem? If so, carry it out, if not, explain why the strategy would not work and move on. $\int sin^2 \theta cos^4 \theta d\theta$ Part 2: Powers of Tangent and Secant In this part of the lab we investigate integrals of the form $\int tan^n \theta sec^m \theta d\theta$. There are two cases that are fairly straightforward that we investigate here. Problem 1 explores one case and problem 2 the second. 1. Consider $\int tan^7 \theta sec^2 \theta d\theta$. Evaluate the integral by making the substitution $w = tan \theta$.

          Lab 4 Trig Integrals
Part 1 Powers of Sine and Cosine The purpose of part one of this activity is to develop a strategy for integrating powers
of Sine and Cosine, that is, integrals of the form $\int sin^n \theta cos^m \theta d\theta$. Work with your neighbor to make sense of the
following problems as you work toward developing your own general strategy.
Would your strategy work for any of these in the table below? Indicate which by writing the simplified form of the
integrand after substitution.
Problem
Substitution
New integral in w
$\int sin^3 \theta cos \theta d\theta$
$w = sin \theta$
$\int w^3 dw$
$\int sin^5 \theta cos \theta d\theta$
$\int sin \theta cos^3 \theta d\theta$
$\int cos^4 \theta d\theta$
Describe the general form of the integrands in the table above.
Briefly describe your strategy for integrating the functions in the table above
1. Evaluate $\int sin^3 \theta cos \theta d\theta$.
2. Now consider $\int sin \theta cos^3 \theta d\theta$.
How is this problem different from the problems on the last page?
Slick Maneuver of the Day: "Pull off" one of the powers of $cos \theta$
$cos^3 \theta = cos^2 \theta cos \theta$
and use the identity $sin^2 \theta + cos^2 \theta = 1$ to transform the problem into one of the type in problem 1. Try it yourself
Would the "slick maneuver of the day" also work for the following integrals? If so, indicate the algebra maneuver, the
trig identity, and the new integral after the substitution.
Problem
Perform Algebra Maneuver
Use Trig Identity
Make Substitution
$\int sin^3 \theta cos \theta d\theta$
$\int sin^3 \theta cos^2 \theta d\theta$
$\int sin^3 \theta (1 - sin^2 \theta) cos \theta d\theta$
$\int w^3 (1 - w^2) dw$
$\int cos^3 \theta d\theta$
$\int sin^5 \theta d\theta$
$\int sin^3 \theta cos^2 \theta d\theta$
Wait a minute, would the strategy outlined above work for the following problem? If so, carry it out, if not, explain why
the strategy would not work and move on.
$\int sin^2 \theta cos^4 \theta d\theta$
Part 2: Powers of Tangent and Secant In this part of the lab we investigate integrals of the form $\int tan^n \theta sec^m \theta d\theta$.
There are two cases that are fairly straightforward that we investigate here. Problem 1 explores one case and problem 2
the second.
1. Consider $\int tan^7 \theta sec^2 \theta d\theta$. Evaluate the integral by making the substitution $w = tan \theta$.

lab 4 trig integrals part 1 powers of sine and cosine the purpose of part one of this activity is to develop a strategy for integrating powers of sine and cosine that is integrals of the for 03377

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