Question

I need to know how to integrate this

Answer 1

use partial fractions here\[\frac{3x+4}{(x^2+4)(x-3)}=\frac{Ax+B}{x^2+4}+\frac{C}{x-3}\]and find A,B and C

Answer 2

Split using partial fractions: \[\frac{ 3x+4 }{ (x ^{2}+4)(x-3) } = \frac{ A }{ x-3 } + \frac{ Bx + C}{ (x ^{2} + 4) } = \frac{ A(x^{2} +4)+(Bx+C)(x-3) }{ (x ^{2}+4)(x-3) }\]
Set equal to 3, looking at just numerator:
\[3*3 + 4 = 13 = A(9+4) + (3B+C)(0) = 13A \rightarrow A = 1\]
Set x equal to 0:
\[0 + 4 = 4 = 4A + C \rightarrow 4 = 4+C \rightarrow C = 0\]
Set x equal to 1:
\[3 + 4 =A(1+4) + (B+C)(1-3) = 5A -2B -2C \rightarrow 5 - 2B - 0 \rightarrow 2 = -2B \rightarrow B = -1\]
Therefore:
\[\frac{ 3x+4 }{ (x ^{2}+4)(x-3)} = \frac{ 1 }{ x-3 } - \frac{ x }{ x ^{2} + 4 }\]
Now integrate those.

Answer 3

thank you very much, you have no idea how much i love you xD

Answer 4

No problemo
The integral of 1/(x-3) is ln(x-3)
and the integral of -x/(x^2 + 4):
u-sub: u = x^2 + 4, du = 2x dx
\[\frac{ -1 }{ 2 } \int\limits_{}^{}\frac{ 1 }{ u }du = \frac{ -1 }{ 2 }\ln(u) = \frac{ -1 }{ 2 }\ln(x ^{2}+4)\]
\[\left( \ln(2-3) - \frac{ 1 }{ 2 }\ln(4+4) \right)-\left( \ln(0-3) - \frac{ 1 }{ 2 }\ln(0+4) \right) \]
\[= \ln(-1)-\frac{ 1 }{ 2 }\ln(8)-\ln(-3)-\frac{ 1 }{ 2 }\ln(4) = \ln(\frac{ 1 }{ 3\sqrt{2} })\]