Skip to main content

`int (x^2 + x + 1)/(x^2 + 1)^2 dx` Evaluate the integral

`int (x^2+x+1)/(x^2+1)^2 dx`


To solve, apply partial fraction decomposition.


To express the integrand as sum of proper rational functions, set the equation as follows:


`(x^2+x+1)/(x^2+1)^2=(Ax+B)/(x^2+1)+(Cx+D)/(x^2+1)^2`


Multiply both sides by the LCD.


`x^2+x+1=(Ax+B)(x^2+1) + Cx + D`


`x^2+x+1=Ax^3+Bx^2+Ax+B+Cx+D`


`x^2+x+1=Ax^3+Bx^2+(A+C)x+B+D`


Express the left side as a polynomial with degree 3.


`0x^3+x^2+x+1=Ax^3+Bx^2+(A+C)x+B+D`


For the two sides to be equal, the two polynomials should be the same. So set the coefficients of the polynomials equal to each other.


x^3:


`0=A`     (Let this be EQ1.)


x^2:


`1=B `     (Let this be EQ2.)


x:


`1=A+C`     (Let this be EQ3.)


Constant:


`1=B+D`     (Let this be EQ4.)


In the equations, the values of A and B are already known. So only the values of C and D have to be solved. To do so, plug-in the value of A to EQ3. 


`1=A+C`


`1=0+C`


`1=C`


Also, plug-in the value of B to EQ4.


`1=B+D`


`1=1+D`


`0=D`


So the partial fraction decomposition of the integrand is:


`(x^2+x+1)/(x^2+1)^2=(0x+1)/(x^2+1)+(1x+0)/(x^2+1)^2=1/(x^2+1)+x/(x^2+1)^2`


Taking the integral of it result to:


`int (x^2+x+1)/(x^2+1)^2 dx`


`= int (1/(x^2+1)+x/(x^2+1)^2) dx`


`= int 1/(x^2+1)dx + int x/(x^2+1)^2dx`


For the first integral, apply the formula int 1/(u^+a^2)=1/a tan^(-1) u/a+C.


For the second integral, apply u-substitution method.


     `u=x^2+1`


     `du=2x dx`


     `(du)/2=xdx`


`= int 1/(x^2+1)dx + int 1/u^2 * (du)/2`


`= int 1/(x^2+1)dx +1/2 int u^(-2) du`


`= tan^(-1)x -1/2u^(-1)+C`


`= tan ^(-1)x - 1/(2u)+C`


Substitute back `u=x^2+1` .


`= tan ^(-1)x - 1/(2(x^2+1))+C`



Therefore, `int (x^2+x+1)/(x^2+1)^2dx= tan ^(-1)x - 1/(2(x^2+1))+C` .

Comments

Post a Comment

Popular posts from this blog

Is there a word/phrase for "unperformant"?

As a software engineer, I need to sometimes describe a piece of code as something that lacks performance or was not written with performance in mind. Example: This kind of coding style leads to unmaintainable and unperformant code. Based on my Google searches, this isn't a real word. What is the correct way to describe this? EDIT My usage of "performance" here is in regard to speed and efficiency. For example, the better the performance of code the faster the application runs. My question and example target the negative definition, which is in reference to preventing inefficient coding practices. Answer This kind of coding style leads to unmaintainable and unperformant code. In my opinion, reads more easily as: This coding style leads to unmaintainable and poorly performing code. The key to well-written documentation and reports lies in ease of understanding. Adding poorly understood words such as performant decreases that ease. In addressing the use of such a poorly ...
Post a Comment
Read more

A man has a garden measuring 84 meters by 56 meters. He divides it into the minimum number of square plots. What is the length of the square plots?

We wish to divide this man's garden into the minimum number of square plots possible. A square has all four sides with the same length.Our garden is a rectangle, so the answer is clearly not 1 square plot. If we choose the wrong length for our squares, we may end up with missing holes or we may not be able to fit our squares inside the garden. So we have 84 meters in one direction and 56 meters in the other direction. When we start dividing the garden in square plots, we are "filling" those lengths in their respective directions. At each direction, there must be an integer number of squares (otherwise, we get holes or we leave the garden), so that all the square plots fill up the garden nicely. Thus, our job here is to find the greatest common divisor of 84 and 56. For this, we prime factor both of them: `56 = 2*2*2*7` `84 = 2*2*3*7` We can see that the prime factors and multiplicities in common are `2*2*7 = 28` . This is the desired length of the square plots. If you wi...
Post a Comment
Read more

What warning does Chuchundra issue to Rikki?

Chuchundra, the sniveling, fearful muskrat who creeps around walls because he is too terrified to go into the center of a room, meets Rikki in the middle of the night. He insults Rikki by begging him not to kill him. He then insults him by suggesting that Nag might mistake Chuchundra for Rikki. He says, "Those who kill snakes get killed by snakes."  He issues this warning to Rikki not to help keep Rikki safe but as a way of explaining why Rikki's presence gives him, Chuchundra, more reason to fear.  Chuchundra starts to tell Rikki what Chua the rat told him--but breaks it off when he realizes he might be overheard by Nag. He says, "Nag is everywhere, Rikki-Tikki." Rikki threatens to bite Chuchundra to get him to talk. Even then, Chuchundra won't overtly reveal any information. But he does say, "Can't you hear, Rikki-Tikki?" This is enough of a clue for the clever mongoose. He listens carefully and can just make out the "faintest scratch-s...
Post a Comment
Read more