What is the integral of ((x+3))/((2x+1)) ?

The integral of ((x+3))/((2x+1)) is 1/2 x+5/4 ln|2x+1|+C

integral of ((x+3))/((2x+1))

integral

\frac{( x + 3 )}{( 2 x + 1 )}

\frac{1}{2} x + \frac{5}{4} ln ∣ 2 x + 1 ∣ + C

Solution steps

\int \frac{x + 3}{2 x + 1} d x

Long division

\frac{x + 3}{2 x + 1} : \frac{1}{2} + \frac{5}{2 ( 2 x + 1 )}

= \int \frac{1}{2} + \frac{5}{2 ( 2 x + 1 )} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{2} d x + \int \frac{5}{2 ( 2 x + 1 )} d x

\int \frac{1}{2} d x = \frac{1}{2} x

\int \frac{5}{2 ( 2 x + 1 )} d x = \frac{5}{4} ln ∣ 2 x + 1 ∣

= \frac{1}{2} x + \frac{5}{4} ln ∣ 2 x + 1 ∣

Add a constant to the solution

= \frac{1}{2} x + \frac{5}{4} ln ∣ 2 x + 1 ∣ + C

in t e g r a l 2 x sin (x^{2} + 1)

in t e g r a l \frac{( x + 3 )}{( x ^{2} - x + 6 )}

in t e g r a l \frac{t ^{2}}{2} + \frac{t ^{3}}{3} + \frac{t ^{4}}{4}

in t e g r a l (\frac{sin ( x )}{x})^{2}

in t e g r a l \frac{- 4}{( x ^{2} + 4 ) ^{2}}

What is the integral of ((x+3))/((2x+1)) ?
The integral of ((x+3))/((2x+1)) is 1/2 x+5/4 ln|2x+1|+C