Question

Calculus 1. Question in the comments

Answer 1

\[\int\limits_{none}^{none} \frac{ 11x+4 }{ (3+6x-x ^{2})^{2} } dx\]

Answer 2

this is supposed to be an indefinite integral.

Answer 3

I cannot easily do u-substitution

Answer 4

Use partial fraction decomposition

\[\frac{ 11x+4 }{(-x^2+6x+3)^2 }=\frac{ Ax+B }{ -x^x+6x+3 }+\frac{ Cx+D }{ (-x^2+6x+3)^2 }\]

Answer 5

Multiply to eliminate fractions
\[11x+4=(Ax+B)(-x^2+6x+3)+Cx+D\]

Answer 6

Would you mind showing that to me? I only learned u-substitution.

Answer 7

you mean -x^2 instand of x^x right?

Answer 8

yes I do mean -x². Read up on it here: http://tutorial.math.lamar.edu/Classes/CalcII/PartialFractions.aspx
There's a table in the middle of the page that shows what format the decomposed fraction is supposed to do. I'll work through this one.

Answer 9

Once you have the fractions eliminated, multiply everything out
\[11x+4=-Ax^3+6Ax^2+3Ax-Bx^2+6Bx+3B+Cx+D\]
Group everything together by the x variable
\[11x+4=x^3(-A)+x^2(6A-B)+x(3A+6B+C)+(3B+D)\]
*check my multiplication please*

Answer 10

checks out.

Answer 11

Now you can equate coefficients to create a system of equations.

On the left, there is no x³ term and on the right the coefficient of x³ is -A, so the equation is
\[0=-A\]
For x², it's
\[0=6A-B\]
For x,
\[11=3A+6B+C\]
For the constant
4 = 3B+D

Answer 12

that makes sense

Answer 13

When I solve the system I get A = 0, B = 0, C = 11, and D = 4, which puts us right where we started :/. PFD is usually a solid method for integrating rational functions. Let me see if I can figure another method

Answer 14

Did you solve the system of equations by going off of the fact that you knew A=0?

Answer 15

yes I did. Ignore this picture for now