What is the integral of (x^2+21)/(x^2+25) ?

The integral of (x^2+21)/(x^2+25) is -4/5 arctan(x/5)+x+C

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integral of (x^2+21)/(x^2+25)

\int \frac{x ^{2} + 21}{x ^{2} + 25} d x

- \frac{4}{5} arctan (\frac{x}{5}) + x + C

Solution steps

\int \frac{x ^{2} + 21}{x ^{2} + 25} d x

\frac{x ^{2} + 21}{x ^{2} + 25} = - \frac{4}{x ^{2} + 25} + 1

= \int - \frac{4}{x ^{2} + 25} + 1 d x

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{4}{x ^{2} + 25} d x + \int 1 d x

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

\int 1 d x = x

= - \frac{4}{5} arctan (\frac{x}{5}) + x

Add a constant to the solution

= - \frac{4}{5} arctan (\frac{x}{5}) + x + C

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What is the integral of (x^2+21)/(x^2+25) ?
The integral of (x^2+21)/(x^2+25) is -4/5 arctan(x/5)+x+C