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Calculus 1 Problems & Solutions – Chapter 6 – Section 6.5.1.4 |
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6.5.1.4
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Review |
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Recall that the derivative of the arcsin function is:
Example 1.1
Calculate:
Solution
EOS
The integrand in the following example isn't the derivative of the arcsin function and can't be transformed into one.
Example 1.2
Compute:
Solution
EOS
Inverse Trigonometric Substitution
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Fig. 1.1
Direct And Inverse Substitutions.
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Example 2.1
Evaluate:
Solution
EOS
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Fig. 2.1
Labelling A Right Triangle According To Substitution.. |
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3. Integrals Involving 1/(a2 + x2) |
Recall that the derivative of the arctan function is:
Example 3.1
Find:
Solution
EOS
The integrand in the following example isn't the derivative of the arctan function and can't be transformed into one.
Example 3.2
Calculate:
Solution
EOS
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Example 4.1
Compute:
Solution
EOS
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5. Before Attempting An Inverse Trigonometric Substitution |
Before attempting to use an inverse trigonometric
substitution, you should examine to see if a direct substitution, which is
simpler, would work. For example, the integral:
can be handled by the direct substitution u = 9 – x2.
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Problems & Solutions |
1. Calculate:
Solution
2. Compute:
Solution
3. Evaluate:
where a > 0.
Solution
4. Let a > 0 be given. Prove that:
Solution
5. Find the area of the shaded region of the circle in the figure below.
Solution
Solution
a.
Consequently:
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