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Chapter 5: Q. 0 (page 463)

Problem Zero: Read the section and make your own summary of the material.

Short Answer

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01

Step 1. Given information.

given,

Read the section and make your own summary of the material.

02

Step 2. Read the section and make your own summary of the material. 

Three Types of Trigonometric Substitutions

Suppose a is a positive real number. The following substitutions can be useful in solving certain integrals:

(a) If x=asin鈦�u, thena2鈭�x2=acos鈦�u.

This substitution is valid for x鈭�[鈭�a,a]andu鈭�鈭�蟺2,蟺2 .

(b) If x=atan鈦�u,thenx2+a2=a2sec2鈦�u.

This substitution is valid for x鈭�(鈭�鈭�,鈭�)andu鈭�鈭�蟺2,蟺2 .

(c) If x=asec鈦�u,thenx2鈭�a2=atan鈦�u,ifx>a鈭�atan鈦�u,ifx<鈭�a

This substitution is valid for x鈭�(鈭�鈭�,鈭�a]鈭�[a,鈭�)andu鈭�0,蟺2鈭�蟺2,蟺.

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Solve the integral:鈭�lnx3dx.

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