What is the integral of (x-3)/(x^2+49) ?

The integral of (x-3)/(x^2+49) is 1/2 ln|x^2+49|-3/7 arctan(x/7)+C

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integral of (x-3)/(x^2+49)

\int \frac{x - 3}{x ^{2} + 49} d x

\frac{1}{2} ln x^{2} + 49 - \frac{3}{7} arctan (\frac{x}{7}) + C

Solution steps

\int \frac{x - 3}{x ^{2} + 49} d x

Expand

\frac{x - 3}{x ^{2} + 49} : \frac{x}{x ^{2} + 49} - \frac{3}{x ^{2} + 49}

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} + 49} d x - \int \frac{3}{x ^{2} + 49} d x

\int \frac{x}{x ^{2} + 49} d x = \frac{1}{2} ln x^{2} + 49

\int \frac{3}{x ^{2} + 49} d x = \frac{3}{7} arctan (\frac{x}{7})

= \frac{1}{2} ln x^{2} + 49 - \frac{3}{7} arctan (\frac{x}{7})

Add a constant to the solution

= \frac{1}{2} ln x^{2} + 49 - \frac{3}{7} arctan (\frac{x}{7}) + C

x \to 0 lim (\frac{x}{arctan ( x )})

\frac{4}{1 + x ^{2}}

f (x) = cos (x) - x sin (x)

\frac{d}{d x} (e^{x} (x^{2} - 3 x + 1))

\int (x^{2} - 2 x)^{3} d x

What is the integral of (x-3)/(x^2+49) ?
The integral of (x-3)/(x^2+49) is 1/2 ln|x^2+49|-3/7 arctan(x/7)+C