009B Sample Final 3, Problem 6

Find the following integrals

(a)  $\int {\frac {3x-1}{2x^{2}-x}}~dx$

(b)  $\int {\frac {\sqrt {x+1}}{x}}~dx$

Foundations:
Through partial fraction decomposition, we can write the fraction
${\frac {1}{(x+1)(x+2)}}={\frac {A}{x+1}}+{\frac {B}{x+2}}$
for some constants $A,B.$

Solution:

(a)

Step 1:
First, we factor the denominator to get
$\int {\frac {3x-1}{2x^{2}-x}}~dx=\int {\frac {3x-1}{x(2x-1)}}.$
We use the method of partial fraction decomposition.
We let
${\frac {3x-1}{x(2x-1)}}={\frac {A}{x}}+{\frac {B}{2x-1}}.$

Step 2:

(b)

Step 1:

Step 2:

Final Answer:
(a)
(b)

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