3
votes
4answers
71 views

Shorter way to integrate $\int \frac{x^9}{(x^2+4)^6} \, \mathrm{d}x$

$$ I=\int \frac{x^9}{(x^2+4)^6}\mathrm{d}x $$ Yeah I know, I can substitute: $$t=x^2+4\text{ or }2\tan\theta$$ So that: $$I=\frac12\int\frac{(t-4)^4}{t^6}\mathrm{d}t\text{ or } ...
2
votes
3answers
153 views

How to evaluate $\int \frac{x^6}{x^4-1} \, \mathrm{d}x.$

Evaluate the integral: $$\int \frac{x^6}{x^4-1} \, \mathrm{d}x$$ After a lot of help I have reached this point: $x^2 = Ax^3 - Ax + Bx^2 - B + Cx^3 + Cx^2 + Cx + C + Dx^3 - Dx^2 + Dx - D$ But now I ...
1
vote
2answers
49 views

Partial Fractions Integration Question

$$\int\frac{x^5+x-1}{x^3 +1} dx$$ Have tried everything ... polynomial long division, partial fractions, trig substitution etc... Not for an assignment, so if a complete solution could be provided ...
0
votes
7answers
67 views

Help with integration using partial fractions

I'm not sure how to get the values for $A$ and $B$ for the expression $$ \frac{3}{x^2 - 16}. $$ I've split the expression into $$ \frac{A}{x - 4} + \frac{B}{x + 4}. $$ I don't know what to do ...
1
vote
1answer
102 views

How to integrate $\int{\frac{6x}{x^3+8}dx}$

I'm having some trouble solving this integral using partial fraction method: $$\int{\frac{6x}{x^3+8}dx}.$$ After expanding $x^3+8$ into $(x-2)(x^2+2x+4)$ and expanding the original integral into ...
4
votes
3answers
196 views

Integral of rational functions.

I want to evaluate this integral: $$\int{\frac{ax+b}{(x^2+2px+q)^n}}dx$$ The book only says to integrate by parts $\int{\dfrac{1}{(x^2+2px+q)^{n-1}}dx}$, for simplicity if $n = 2$ I get: ...
3
votes
1answer
272 views

Integrating a partial fraction with multiple quadratic denominators

When integrating a real rational fraction, you first break it into partial fractions. You then end up with fractions with linear denominators $\frac{A}{(x-b)^n}$, which are easy. You also end up with ...
3
votes
3answers
1k views

Integral of $\int \frac{x^2 - 5x + 16}{(2x+1)(x-2)^2}dx$

I am trying to find the integral of this by using integration of rational functions by partial fractions. $$\int \frac{x^2 - 5x + 16}{(2x+1)(x-2)^2}dx$$ I am not really sure how to start this but ...
3
votes
2answers
1k views

Integrate using Partial Fraction decomposition, completing the square

The given problem is $\int{x\over x^3-1}dx$. I know this equals $${1\over3}\int {1\over x-1}-{x-1\over x^2+x+1}dx,$$ which can be separated into $${1\over3}\int {1\over x-1}dx - ...