What is the integral of (5x^2+2x+29)/((x-3)(x^2+7)) ?

The integral of (5x^2+2x+29)/((x-3)(x^2+7)) is 5ln|x-3|+2/(sqrt(7))arctan(x/(sqrt(7)))+C

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integral of (5x^2+2x+29)/((x-3)(x^2+7))

\int \frac{5 x ^{2} + 2 x + 29}{( x - 3 ) ( x ^{2} + 7 )} d x

5 ln ∣ x - 3 ∣ + \frac{2}{7} arctan (\frac{x}{7}) + C

Solution steps

\int \frac{5 x ^{2} + 2 x + 29}{( x - 3 ) ( x ^{2} + 7 )} d x

Expand

\frac{5 x ^{2} + 2 x + 29}{( x - 3 ) ( x ^{2} + 7 )} : \frac{5 x ^{2}}{( x - 3 ) ( x ^{2} + 7 )} + \frac{2 x}{( x - 3 ) ( x ^{2} + 7 )} + \frac{29}{( x - 3 ) ( x ^{2} + 7 )}

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 x ^{2}}{( x - 3 ) ( x ^{2} + 7 )} d x + \int \frac{2 x}{( x - 3 ) ( x ^{2} + 7 )} d x + \int \frac{29}{( x - 3 ) ( x ^{2} + 7 )} d x

\int \frac{5 x ^{2}}{( x - 3 ) ( x ^{2} + 7 )} d x = 5 (\frac{9}{16} ln ∣ x - 3 ∣ + \frac{1}{16} (\frac{7}{2} ln x^{2} + 7 + 37 arctan (\frac{x}{7})))

\int \frac{2 x}{( x - 3 ) ( x ^{2} + 7 )} d x = 2 (\frac{3}{16} ln ∣ x - 3 ∣ + \frac{1}{16} (- \frac{3}{2} ln x^{2} + 7 + 7 arctan (\frac{x}{7})))

\int \frac{29}{( x - 3 ) ( x ^{2} + 7 )} d x = 29 (\frac{1}{16} ln ∣ x - 3 ∣ + \frac{1}{16} (- \frac{1}{2} ln x^{2} + 7 - \frac{3}{7} arctan (\frac{x}{7})))

= 5 (\frac{9}{16} ln ∣ x - 3 ∣ + \frac{1}{16} (\frac{7}{2} ln x^{2} + 7 + 37 arctan (\frac{x}{7}))) + 2 (\frac{3}{16} ln ∣ x - 3 ∣ + \frac{1}{16} (- \frac{3}{2} ln x^{2} + 7 + 7 arctan (\frac{x}{7}))) + 29 (\frac{1}{16} ln ∣ x - 3 ∣ + \frac{1}{16} (- \frac{1}{2} ln x^{2} + 7 - \frac{3}{7} arctan (\frac{x}{7})))

Simplify

5 (\frac{9}{16} ln ∣ x - 3 ∣ + \frac{1}{16} (\frac{7}{2} ln x^{2} + 7 + 37 arctan (\frac{x}{7}))) + 2 (\frac{3}{16} ln ∣ x - 3 ∣ + \frac{1}{16} (- \frac{3}{2} ln x^{2} + 7 + 7 arctan (\frac{x}{7}))) + 29 (\frac{1}{16} ln ∣ x - 3 ∣ + \frac{1}{16} (- \frac{1}{2} ln x^{2} + 7 - \frac{3}{7} arctan (\frac{x}{7}))) : 5 ln ∣ x - 3 ∣ + \frac{2}{7} arctan (\frac{x}{7})

= 5 ln ∣ x - 3 ∣ + \frac{2}{7} arctan (\frac{x}{7})

Add a constant to the solution

= 5 ln ∣ x - 3 ∣ + \frac{2}{7} arctan (\frac{x}{7}) + C

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What is the integral of (5x^2+2x+29)/((x-3)(x^2+7)) ?
The integral of (5x^2+2x+29)/((x-3)(x^2+7)) is 5ln|x-3|+2/(sqrt(7))arctan(x/(sqrt(7)))+C