What is the integral of (4x+3)/(2x+5) ?

The integral of (4x+3)/(2x+5) is 2x-7/2 ln|2x+5|+C

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integral of (4x+3)/(2x+5)

\int \frac{4 x + 3}{2 x + 5} d x

2 x - \frac{7}{2} ln ∣ 2 x + 5 ∣ + C

Solution steps

\int \frac{4 x + 3}{2 x + 5} d x

Long division

\frac{4 x + 3}{2 x + 5} : 2 - \frac{7}{2 x + 5}

= \int 2 - \frac{7}{2 x + 5} 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 2 d x - \int \frac{7}{2 x + 5} d x

\int 2 d x = 2 x

\int \frac{7}{2 x + 5} d x = \frac{7}{2} ln ∣ 2 x + 5 ∣

= 2 x - \frac{7}{2} ln ∣ 2 x + 5 ∣

Add a constant to the solution

= 2 x - \frac{7}{2} ln ∣ 2 x + 5 ∣ + C

\int \frac{sin ( 2 x )}{( 1 + cos ( 2 x ) ) ^{5}} d x

\int x^{2} - 2 x + 10 d x

\int 3 x^{5} y^{3} d y

\int \frac{x}{cos ^{2} ( 4 x ^{2} + \frac{1}{a} )} d x

\int \frac{3 t}{2 t ^{2} + 5 t + 2} d t

What is the integral of (4x+3)/(2x+5) ?
The integral of (4x+3)/(2x+5) is 2x-7/2 ln|2x+5|+C