SOLUTION: Please help me solve this equation: Evaluate the following integrals using trigonometric substitutions. a) {{{ int ( x^2 / ( sqrt (1-(x^2) ) ), dx ) }}} b) {{{ int ( 1/ (sqrt (

Algebra ->  Test -> SOLUTION: Please help me solve this equation: Evaluate the following integrals using trigonometric substitutions. a) {{{ int ( x^2 / ( sqrt (1-(x^2) ) ), dx ) }}} b) {{{ int ( 1/ (sqrt (      Log On


   

Question 355862: Please help me solve this equation:
Evaluate the following integrals using trigonometric substitutions.
a) +int+%28+x%5E2+%2F+%28+sqrt+%281-%28x%5E2%29+%29+%29%2C+dx+%29+
b) +int+%28+1%2F+%28sqrt+%2829-4x%2B%28x%5E2%29%29+%29%2C+dx+%29+
Found 2 solutions by Fombitz, jim_thompson5910:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Use a substitution.
Let x=sin%28u%29, dx=cos%28u%29du
x%5E2=%28sin%28u%29%29%5E2
1-x%5E2=1-%28sin%28u%29%29%5E2=%28cos%28u%29%29%5E2
sqrt%281-x%5E2%29=cos%28u%29
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Take it from there and solve for the integral in u, then back substitute to x.
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Complete the square in the denominator.


Substitute, let u=x-2, du=dx

Substitute again, let u=5tan%28v%29, du=5%28sec%28v%29%5E2%29dv

sqrt%28u%5E2%2B25%29=5%2Asec%28v%29
1%2Fsqrt%28u%5E2%2B25%29=1%2F%285%2Asec%28v%29%29

+int+%28+1%2F+%28sqrt+%2829-4x%2B%28x%5E2%29%29+%29%2C+dx+%29+=int+%28+sec%28v%29%2C+dv%29+
Take it from here and get the solution in v, back substitute for u, and then back substitute for x.
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Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
a) Hint: Draw a triangle with legs of x, sqrt%281-x%5E2%29 and with a hypotenuse of 1. So sin%28theta%29=x%2F1=x ('x' is the leg opposite the angle theta). So because x=sin%28theta%29,


So the integral then transforms to





But we know that , so...





and then the cosine terms cancel to leave you with





Let me know if this is enough to get you started.