009B Sample Midterm 3, Problem 5 Detailed Solution
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Evaluate the indefinite and definite integrals.
(a)
(b)
Background Information: |
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1. Integration by parts tells us that |
2. Recall the trig identity |
Solution:
(a)
Step 1: |
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To evaluate this integral, we use integration by parts. |
Let and |
Then, and |
Step 2: |
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Using integration by parts, we get |
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(b)
Step 1: |
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One of the double angle formulas is |
Solving for we get |
Plugging this identity into our integral, we get |
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Step 2: |
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If we integrate the first integral, we get |
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Step 3: |
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For the remaining integral, we need to use -substitution. |
Let |
Then, and |
Also, since this is a definite integral and we are using -substitution, |
we need to change the bounds of integration. |
We have and |
So, the integral becomes |
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Final Answer: |
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(a) |
(b) |