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

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

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

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

4 arctan (x) + 4 ln 1 + x^{2} + C

Solution steps

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

Expand

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

\int \frac{4}{1 + x ^{2}} d x = 4 arctan (x)

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

= 4 arctan (x) + 4 ln 1 + x^{2}

Add a constant to the solution

= 4 arctan (x) + 4 ln 1 + x^{2} + C

\int 9 x^{2} + 4 x - 3 d x

\frac{d}{d x} (\frac{2 x - x ^{2}}{x} - 3 cot (x))

x \to 4 lim (2 x^{2} + 7)

\frac{d h}{d t} = \frac{( 7 - e ^{- \frac{t}{5}} )}{6} - \frac{2}{9} h

2 x y^{3} = x y + 6

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