How do you integrate #int (3x^2 - 1) /( ((x^2)+2) (x-3))# using partial fractions?

Question

Evelyn Agard · Accepted Answer

#26/11ln|x-3|+7/22ln(x^2+2)+21/(11sqrt2)tan^(-1)(x/sqrt2)+K#.

We split the Integrand as : #(3x^2-1)/((x^2+2)(x-3))=A/(x-3)+(Bx+C)/(x^2+2),......................(star)# where, #A,B,C in RR#. #A# can be quickly determined by Heavyside"s Cover-up Method as #A=[(3x^2-1)/(x^2+2)]_(x=3)=26/11# Simplifying #(star)# & comparing rational polys. on both sides, we get, #A(x^2+2)+(Bx+C)(x-3)=3x^2-1.....................(1)# Taking #x=0, &, A=26/11# in #(1)#, #26/11(2)+C(-3)=-1rArrC=21/11#. #(1), x=1, A=26/11, C=21/11# #rArr26/11(3)+(B+21/11)(-2)=2rArrB=7/11#. With these #A,B,C#, we have, #int(3x^2-1)/((x^2+2)(x-3))dx#, #=26/11int1/(x-3)dx+1/11int(7x+21)/(x^2+2)dx#, #=26/11ln|x-3|+7/11*1/2int(2x)/(x^2+2)dx+21/11int1/(x^2+2)dx#, #=26/11ln|x-3|+7/22int(d(x^2+2))/(x^2+2)+21/11*1/sqrt2*tan^(-1)(x/sqrt2)#, #=26/11ln|x-3|+7/22ln(x^2+2)+21/(11sqrt2)tan^(-1)(x/sqrt2)+K#. Have fun with math!

Emily Ammerman · Answer

undefined To integrate $\frac{{3x^2 - 1}}{{(x^2 + 2)(x - 3)}}$ using partial fractions:

1. Factor the denominator $x^2 + 2$ as irreducible quadratic, and $x - 3$ as linear.
2. Write the fraction in the form of partial fractions: $\frac{{3x^2 - 1}}{{(x^2 + 2)(x - 3)}} = \frac{{Ax + B}}{{x^2 + 2}} + \frac{{C}}{{x - 3}}$.
3. Clear the denominators and simplify to solve for the unknown coefficients A, B, and C.
4. Once you have found A, B, and C, integrate each term separately.
5. Finally, sum up the integrals of each partial fraction.

Following these steps:

1. The factorization of the denominator gives $x^2 + 2 = (x + i\sqrt{2})(x - i\sqrt{2})$.
2. The partial fraction decomposition becomes: $\frac{{3x^2 - 1}}{{(x + i\sqrt{2})(x - i\sqrt{2})(x - 3)}} = \frac{{Ax + B}}{{x + i\sqrt{2}}} + \frac{{Cx + D}}{{x - i\sqrt{2}}} + \frac{{E}}{{x - 3}}$.
3. Clear the denominators and equate coefficients to find A, B, C, D, and E.
4. After solving for A, B, C, D, and E, integrate each partial fraction term.
5. Finally, sum up the integrals to get the result.

This process can be somewhat complex due to the presence of complex roots. However, once you've found the constants, integrating each term individually becomes straightforward.