What is the integral of (x-4)/(x^2+25) ?

The integral of (x-4)/(x^2+25) is 1/2 ln|x^2+25|-4/5 arctan(x/5)+C

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integral of (x-4)/(x^2+25)

\int \frac{x - 4}{x ^{2} + 25} d x

\frac{1}{2} ln x^{2} + 25 - \frac{4}{5} arctan (\frac{x}{5}) + C

Solution steps

\int \frac{x - 4}{x ^{2} + 25} d x

Expand

\frac{x - 4}{x ^{2} + 25} : \frac{x}{x ^{2} + 25} - \frac{4}{x ^{2} + 25}

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} + 25} d x - \int \frac{4}{x ^{2} + 25} d x

\int \frac{x}{x ^{2} + 25} d x = \frac{1}{2} ln x^{2} + 25

\int \frac{4}{x ^{2} + 25} d x = \frac{4}{5} arctan (\frac{x}{5})

= \frac{1}{2} ln x^{2} + 25 - \frac{4}{5} arctan (\frac{x}{5})

Add a constant to the solution

= \frac{1}{2} ln x^{2} + 25 - \frac{4}{5} arctan (\frac{x}{5}) + C

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What is the integral of (x-4)/(x^2+25) ?
The integral of (x-4)/(x^2+25) is 1/2 ln|x^2+25|-4/5 arctan(x/5)+C