What is the integral of (x+5)/(x^2+25) ?

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

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integral of (x+5)/(x^2+25)

\int \frac{x + 5}{x ^{2} + 25} d x

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

Solution steps

\int \frac{x + 5}{x ^{2} + 25} d x

Expand

\frac{x + 5}{x ^{2} + 25} : \frac{x}{x ^{2} + 25} + \frac{5}{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{5}{x ^{2} + 25} d x

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

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

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

Add a constant to the solution

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

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\int x^{3} (4 x^{2} + 5) d x

\int \frac{1}{x ^{\frac{9}{2}}} d x

\frac{6 ( s + 34 )}{s ( s ^{2} + 10 s + 34 )}

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