Expand the integral $\int\left(\frac{x}{4x^2+2x}+\frac{-3}{2x+1}\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
$\int\frac{x}{4x^2+2x}dx+\int\frac{-3}{2x+1}dx$
Intermediate steps
3
The integral $\int\frac{-3}{2x+1}dx$ results in: $-\frac{3}{2}\ln\left(2x+1\right)$
Combining like terms $\frac{1}{4}\ln\left(2x+1\right)$ and $-\frac{3}{2}\ln\left(2x+1\right)$
$-\frac{5}{4}\ln\left(2x+1\right)$
9
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
$-\frac{5}{4}\ln\left(2x+1\right)+C_0$
Final answer to the problem
$-\frac{5}{4}\ln\left(2x+1\right)+C_0$
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Integration assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.