Question

Determine S(1+ 4/3x-1
+ 3/x+2) dx by partial fractions
​

Answer 1

`\large\boxed{\int\left(1+(4)/(3x-1)+(3)/(x+2)\right)\ dx=x+(4)/(3)\ln(3x-1)+3\ln(x+2)}`

Explanation:

`\large{\int}\\ormal\left(1+(4)/(3x-1)+(3)/(x+2)\right)\ dx=\int1\ dx+\int(4)/(3x-1)\ dx+\int(3)/(x+2)\ dx\\\\(1)\int1\ dx=x\\\\(2)\int(4)/(3x-1)\ dx\Rightarrow\left|\begin{array}{ccc}3x-1=t\\3dx=dt\\dx=(1)/(3)dt\end{array}\right|\Rightarrow\int(4)/(3t)\ dt=(4)/(3)\int(1)/(t)\ dt=(4)/(3)\ln(t)=(4)/(3)\ln(3x-1)\\\\(3)\int(3)/(x+2)\ dx\Rightarrow\left|\begin{array}{ccc}x+2=u\\dx=du\end{array}\right|\Rightarrow\int(3)/(t)\ dt=3\int(1)/(t)\ dt=3\ln(t)=3\ln(x+2)`

`\Downarrow\\\\\int\left(1+(4)/(3x-1)+(3)/(x+2)\right)\ dx=x+(4)/(3)\ln(3x-1)+3\ln(x+2)`