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

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

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

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

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

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 3 \cdot \int \frac{x + a}{x ^{2} + a ^{2}} d x

Expand

\frac{x + a}{x ^{2} + a ^{2}} : \frac{x}{x ^{2} + a ^{2}} + \frac{a}{x ^{2} + a ^{2}}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - 3 (\int \frac{x}{x ^{2} + a ^{2}} d x + \int \frac{a}{x ^{2} + a ^{2}} d x)

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

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

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

Add a constant to the solution

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

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