What is the integral of (10/9 x+14)/(x^2+9) ?

The integral of (10/9 x+14)/(x^2+9) is 5/9 ln|x^2+9|+14/3 arctan(x/3)+C

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integral of (10/9 x+14)/(x^2+9)

\int \frac{\frac{10}{9} x + 14}{x ^{2} + 9} d x

\frac{5}{9} ln x^{2} + 9 + \frac{14}{3} arctan (\frac{x}{3}) + C

Solution steps

\int \frac{\frac{10}{9} x + 14}{x ^{2} + 9} d x

Expand

\frac{\frac{10}{9} x + 14}{x ^{2} + 9} : \frac{\frac{10}{9} x}{x ^{2} + 9} + \frac{14}{x ^{2} + 9}

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{\frac{10}{9} x}{x ^{2} + 9} d x + \int \frac{14}{x ^{2} + 9} d x

\int \frac{\frac{10}{9} x}{x ^{2} + 9} d x = \frac{5}{9} ln x^{2} + 9

\int \frac{14}{x ^{2} + 9} d x = \frac{14}{3} arctan (\frac{x}{3})

= \frac{5}{9} ln x^{2} + 9 + \frac{14}{3} arctan (\frac{x}{3})

Add a constant to the solution

= \frac{5}{9} ln x^{2} + 9 + \frac{14}{3} arctan (\frac{x}{3}) + C

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x \to + 0 lim (\frac{∣ x - 3 ∣}{x - 3})

\int \frac{y ^{2} + 28 y + 196}{( y ^{2} + 196 ) ^{2}} d y

x \to - 4 lim (\frac{x ^{2} - 1}{4 - x})

\int_{0}^{1} cos (\frac{π x}{2}) d x

What is the integral of (10/9 x+14)/(x^2+9) ?
The integral of (10/9 x+14)/(x^2+9) is 5/9 ln|x^2+9|+14/3 arctan(x/3)+C