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

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

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

\int \frac{x + 3}{( x - 2 ) ( x ^{2} + 1 )} d x

ln ∣ x - 2 ∣ - \frac{1}{2} ln x^{2} + 1 - arctan (x) + C

Solution steps

\int \frac{x + 3}{( x - 2 ) ( x ^{2} + 1 )} d x

Take the partial fraction of

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

= \int \frac{1}{x - 2} + \frac{- x - 1}{x ^{2} + 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}{x - 2} d x + \int \frac{- x - 1}{x ^{2} + 1} d x

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

\int \frac{- x - 1}{x ^{2} + 1} d x = - \frac{1}{2} ln x^{2} + 1 - arctan (x)

= ln ∣ x - 2 ∣ - \frac{1}{2} ln x^{2} + 1 - arctan (x)

Add a constant to the solution

= ln ∣ x - 2 ∣ - \frac{1}{2} ln x^{2} + 1 - arctan (x) + C

\int \frac{x ^{3} ln ( x )}{3} d x

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

\int \frac{( 1 - x ) ^{2}}{2 x} d x

\int (- x^{3} + 6 x^{2} - 9 x) d x

\int (\frac{tan ( ln ( x ) )}{x}) d x

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