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

The integral of (x-3)/(x^2+7) is 1/2 ln|x^2+7|-3/(sqrt(7))arctan(x/(sqrt(7)))+C

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

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

\frac{1}{2} ln x^{2} + 7 - \frac{3}{7} arctan (\frac{x}{7}) + C

Solution steps

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

Expand

\frac{x - 3}{x ^{2} + 7} : \frac{x}{x ^{2} + 7} - \frac{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{x}{x ^{2} + 7} d x - \int \frac{3}{x ^{2} + 7} d x

\int \frac{x}{x ^{2} + 7} d x = \frac{1}{2} ln x^{2} + 7

\int \frac{3}{x ^{2} + 7} d x = \frac{3}{7} arctan (\frac{x}{7})

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

Add a constant to the solution

= \frac{1}{2} ln x^{2} + 7 - \frac{3}{7} arctan (\frac{x}{7}) + C

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What is the integral of (x-3)/(x^2+7) ?
The integral of (x-3)/(x^2+7) is 1/2 ln|x^2+7|-3/(sqrt(7))arctan(x/(sqrt(7)))+C