What is the integral of (-46x+158)/((x+2)(x-3)^2) ?

The integral of (-46x+158)/((x+2)(x-3)^2) is 10ln|x+2|-10ln|x-3|-4/(x-3)+C

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integral of (-46x+158)/((x+2)(x-3)^2)

\int \frac{- 46 x + 158}{( x + 2 ) ( x - 3 ) ^{2}} d x

10 ln ∣ x + 2 ∣ - 10 ln ∣ x - 3 ∣ - \frac{4}{x - 3} + C

Solution steps

\int \frac{- 46 x + 158}{( x + 2 ) ( x - 3 ) ^{2}} d x

Take the partial fraction of

\frac{- 46 x + 158}{( x + 2 ) ( x - 3 ) ^{2}} : \frac{10}{x + 2} - \frac{10}{x - 3} + \frac{4}{( x - 3 ) ^{2}}

= \int \frac{10}{x + 2} - \frac{10}{x - 3} + \frac{4}{( x - 3 ) ^{2}} 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{10}{x + 2} d x - \int \frac{10}{x - 3} d x + \int \frac{4}{( x - 3 ) ^{2}} d x

\int \frac{10}{x + 2} d x = 10 ln ∣ x + 2 ∣

\int \frac{10}{x - 3} d x = 10 ln ∣ x - 3 ∣

\int \frac{4}{( x - 3 ) ^{2}} d x = - \frac{4}{x - 3}

= 10 ln ∣ x + 2 ∣ - 10 ln ∣ x - 3 ∣ - \frac{4}{x - 3}

Add a constant to the solution

= 10 ln ∣ x + 2 ∣ - 10 ln ∣ x - 3 ∣ - \frac{4}{x - 3} + C

2 3 x

9 y^{^{''}} + y^{^{'}} = 0

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

\frac{\partial}{\partial x} (y ln (x^{2} + y^{2} + 3) - 6)

\frac{\partial}{\partial y} (3 x cos (y))

What is the integral of (-46x+158)/((x+2)(x-3)^2) ?
The integral of (-46x+158)/((x+2)(x-3)^2) is 10ln|x+2|-10ln|x-3|-4/(x-3)+C