What is the integral of (3x+27)/((2x-2)(x+5)) ?

The integral of (3x+27)/((2x-2)(x+5)) is 5/2 ln|x-1|-ln|x+5|+C

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integral of (3x+27)/((2x-2)(x+5))

\int \frac{3 x + 27}{( 2 x - 2 ) ( x + 5 )} d x

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

Solution steps

\int \frac{3 x + 27}{( 2 x - 2 ) ( x + 5 )} d x

Take the partial fraction of

\frac{3 x + 27}{( 2 x - 2 ) ( x + 5 )} : \frac{5}{2 ( x - 1 )} - \frac{1}{x + 5}

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

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

\int \frac{1}{x + 5} d x = ln ∣ x + 5 ∣

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

Add a constant to the solution

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

y = \frac{2}{x}, y = \frac{1}{2} x, y = 8 x

u^{\frac{3}{2}} (u + c e^{u})

\int_{1}^{\infty} \frac{3}{x ^{3}} d x

\frac{\partial}{\partial x} (x^{3} - x^{2} + y^{3} - y^{2})

\int x e^{x^{2} - 1} d x

What is the integral of (3x+27)/((2x-2)(x+5)) ?
The integral of (3x+27)/((2x-2)(x+5)) is 5/2 ln|x-1|-ln|x+5|+C