Calculus Of One Real Variable By Pheng Kim Ving
Chapter 10: Techniques Of Integration Section 10.5: The Inverse Trigonometric Substitution

 

10.5
The Inverse Trigonometric Substitution

 

 

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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

 

 

Fig. 1.1

 

Direct And Inverse Substitutions.

 

 

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Example 2.1

 

Evaluate:

 

 

Solution

 

 


EOS

 

 

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

 

 

 

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2. Compute:

 

 

Solution

 

 

 

 

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3. Evaluate:

 

 

where a > 0.

 

Solution

 

 

 

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4. Let a > 0 be given. Prove that:

 

 

Solution


 

If x > a then:

 

 

where C = C1 ln a.

 

If x < a then let t = x, so that t > a and dt = dx, then:

 


where C = C2 ln a2.

 

 

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5. Find the area of the shaded region of the circle in the figure below.

 

 

Solution

 

 

 

 

 

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6.

 

Solution

 

a.

 

 

 

Consequently:

 

 

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