(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Here's the question: int(1/(x(x^2+3)*sqrt(1-x^2)))dx

2. Relevant equations

According to the textbook, the answer should be: (1/3)*ln((1-sqrt(1-x^2))/x)+(1/12)*ln((2+sqrt(1-x^2))/(x^2+3))+C

3. The attempt at a solution

1) let t = sqrt(1-x^2), so dt = (-x)/sqrt(1-x^2) dx

2) substituted t and dt into the equation and got the following: -int(1/((1-t^2)(4-t^2)))dt

3) expanded the denominator into (1+t)(1-t)(2+t)(2-t) and used partial fractions to find A,B,C and D in the numerator (respectively)

Using partial fractions, I got: A & B = 1/6, C & D = -1/12

4) From there, I integrated all four portions separately to yield:

-(1/6)*ln(1+t)-(1/6)*ln(1-t)+(1/12)*ln(2+t)+(1/12)*ln(2-t)+C

5) Substituting t = sqrt(1-x^2) back into the equation and collecting like terms, I got:

-(1/6)*ln((1+sqrt(1-x^2))(1-sqrt(1-x^2)))+(1/12)*ln((2+sqrt(1-x^2))(2-sqrt(1-x^2)))+C

6) Finally, multiplied everything inside both ln's and brought out the 2 in the first term (ln(x^2)) to get 1/3:

My final answer: -(1/3)*ln(x)+(1/12)*ln(x^2+3)+c

Thank you for your help in advance!!!!!!

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Integration of Rational Function - problem

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