MATH 155 SUPPLEMENTAL NOTES 8
DERIVATIVES OF INVERSE TRIG FUNCTIONS AND RELATED INTEGRALS
DERIVATIVES OF INVERSE TRIG FUNCTIONS
BASIC FACTS
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Know them, use them, they are your friends!
JEXAMPLE 1:
SOLUTION:
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EXAMPLE 2:
SOLUTION:
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EXAMPLE 3:
SOLUTION:
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EXAMPLE 4:
SOLUTION:
EXAMPLE 5:
SOLUTION:
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EXAMPLE 6:
SOLUTION:
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RELATED INTEGRALS
INTEGRATION FORMULAS
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Valid for u 2 < a 2. |
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Valid for all u. |
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Valid for u 2 < a 2. |
Know them, use them, they are your friends!
JEXAMPLE 7:
SOLUTION:
In this integral, a = 1 and u = x. It is in the form a 2 - u 2, so I will use the sine inverse integral.
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Here is the reference triangle for sin -1(1/2) |
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EXAMPLE 8:
SOLUTION:
For this integral a = 2 and u = 3x, so it is of the form u 2 + a 2. Therefore it is an inverse tangent integral. If u = 3x, then du = 3dx or (1/3)du = dx.
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EXAMPLE 9:
SOLUTION:
In this integral, u = x and a = 5. It is of the form u 2 - a 2, so it is an inverse secant integral.
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EXAMPLE 10:
SOLUTION:
Let a = 1 and u = e x for this integral, therefore it is in the form a 2 + u2 which is the inverse tangent integral. If u = e x, then du = e xdx.
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EXAMPLE 11:
SOLUTION:
Let u = ln x and a = 1 for this integral, therefore this is in the form of the inverse tangent integral. Furthermore, when u = ln x, then du = (1/x)dx.
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EXAMPLE 12:
SOLUTION:
Notice that this integral is in the form of the inverse sine integral, so we will let u = ln x. We will also have to us the method of u-substitution on this one. I have been using this method on the previous examples. This method should be very familiar to you, and it will not be the last time that you will ever use it.
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u = ln x |
When x = 1, then u = 0, and when x = e, then u = 1. |
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EXAMPLE 13:
SOLUTION:
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When x = 1, then u = 1. |
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I have provided you with a variety of worked examples. When working through these examples, please make note of the patterns. The sooner you can recognize what pattern goes with what derivative or integral, the better you will do. Again, if you have any questions or problems with any of these examples, feel free to contact me.
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