Chapter 7, Section 4

Trigonometric Substitution 

 

Trigonometric Substitutions (a > 0)    

   

For integrals involving (a^2 - u^2)^(1/2)let   u = a sin

Then (a^2 - u^2)^(1/2) = a cos , 

where ( -pi/2 <= <= pi/2 )

 

See chart on page 506 for geometric drawing

 

  

For integrals involving (a^2 + u^2)^(1/2)let   u = a tan

Then (a^2 + u^2)^(1/2) = a sec

where ( -pi/2 <= <= pi/2 )

See chart on page 506 for geometric drawing

 

    

For integrals involving (u^2 - a^2)^(1/2)let   u = a sec

Then (u^2 - a^2)^(1/2)  = (+ or -)a tan

where (0<= < pi/2   or  pi/2 <=   <=  pi )

Use the positive value if u > a

and

Use the negative value if u < -a.

See chart on page 506 for geometric drawing

 

 
   

 

 

 

Theorem 7.2  Special Integration Formulas (a >0)

1.) ∫ sqrt(a^2 - u^2)du = 1/2 * [a^2 arcsin (u/a) + u  sqrt(a^2 - u^2)] + C

2.)  ∫ sqrt(u^2 - a^2)du = 1/2 * [u * sqrt(u^2 - a^2) - a^2 Ln| u +  sqrt(u^2 - a^2)|] + C

3.)  ∫ sqrt(u^2 + a^2)du = 1/2 * [u * sqrt(u^2 + a^2) + a^2 Ln| u +  sqrt(u^2 + a^2)|] + C

Finding arc Length (formula from section 6.4)

s = (lower limit = a , upper limit = b) sqrt(1 + [f '(x)]^2) dx 

 
Comparing 2 Fluid Forces ( General equation Section 6.7)

s = (lower limit = c , upper limit = d) h(y)*L(y) dy 

 
   
   
   

More links to help on Visual Calc HP: 

Tutorial on integration using the method of substitution.

 


 

Link to Many Tables on S.O.S Math

 

Contact Ann

 

Ann trying to understand this.