## 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

Go To Problems & Solutions 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. ### Example 2.1

Evaluate: Solution   EOS  # Fig. 2.1

Labelling A Right Triangle According To Substitution..

 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  ### Example 4.1

Compute: Solution   EOS 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.

 Problems & Solutions

1. Calculate: Solution   2. Compute: Solution    3. Evaluate: where a > 0.

Solution   4. Let a > 0 be given. Prove that: Solution If x > a then: where C = C1ln a.

If x < – a then let t = – x, so that t > a and dt = – dx, then: where C = C2ln a2.  5. Find the area of the shaded region of the circle in the figure below. Solution     6. Solution

a.  Consequently:  