How do you integrate $\int \frac{3 x + 5}{x - 2}$ $int (3x+5)/(x-2)$ ?

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How do you integrate $\int \frac{3 x + 5}{x - 2}$ $int (3x+5)/(x-2)$ ?

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Answer 1 · 2017-01-14T03:50:37

$3 x + 11 ln (| x - 2 |) + C$ $3x+11ln(abs(x-2))+C$

Explanation:

First simplify the integrand. This can be done by performing long division, or by my preferred method:

$\int \frac{3 x + 5}{x - 2} d x = \int \frac{3 (x - 2) + 11}{x - 2} d x$ $int(3x+5)/(x-2)dx=int(3(x-2)+11)/(x-2)dx$

$\int \frac{3 x + 5}{x - 2} d x = \int \frac{3 (x - 2)}{x - 2} d x + \int \frac{11}{x - 2} d x$ $color(white)(int(3x+5)/(x-2)dx)=int(3(x-2))/(x-2)dx+int11/(x-2)dx$

$\int \frac{3 x + 5}{x - 2} d x = 3 \int d x + 11 \int \frac{d x}{x - 2}$ $color(white)(int(3x+5)/(x-2)dx)=3intdx+11intdx/(x-2)$

Both of these are simple integration problems. If you're stuck on the second one, try the substitution $u = x - 2 \Rightarrow d u = d x$ $u=x-2=>du=dx$ and recall that $\int \frac{d u}{u} = ln (| u |) + C$ $int(du)/u=ln(absu)+C$ .

$\int \frac{3 x + 5}{x - 2} d x = 3 x + 11 ln (| x - 2 |) + C$ $color(white)(int(3x+5)/(x-2)dx)=3x+11ln(abs(x-2))+C$

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How do you integrate $\int \frac{3 x + 5}{x - 2}$ $int (3x+5)/(x-2)$ ?

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