What is the integral of (8+4x)/(1+x^2) ?

The integral of (8+4x)/(1+x^2) is 8arctan(x)+2ln|1+x^2|+C

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integral of (8+4x)/(1+x^2)

\int \frac{8 + 4 x}{1 + x ^{2}} d x

8 arctan (x) + 2 ln 1 + x^{2} + C

Solution steps

\int \frac{8 + 4 x}{1 + x ^{2}} d x

Expand

\frac{8 + 4 x}{1 + x ^{2}} : \frac{8}{1 + x ^{2}} + \frac{4 x}{1 + x ^{2}}

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{8}{1 + x ^{2}} d x + \int \frac{4 x}{1 + x ^{2}} d x

\int \frac{8}{1 + x ^{2}} d x = 8 arctan (x)

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

= 8 arctan (x) + 2 ln 1 + x^{2}

Add a constant to the solution

= 8 arctan (x) + 2 ln 1 + x^{2} + C

f^{^{'}} (x) = 3.52 x^{2} - 30.1 x + 81

\frac{10}{x}

\frac{d}{d x} (e^{x + 8})

\int sin^{7} (x) cos^{5} (x) d x

\frac{\partial ^{2}}{\partial x \partial y} (\frac{1}{x y})

What is the integral of (8+4x)/(1+x^2) ?
The integral of (8+4x)/(1+x^2) is 8arctan(x)+2ln|1+x^2|+C