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

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

Popular >

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

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

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

Solution steps

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

Expand

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

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

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

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

Add a constant to the solution

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

\frac{d}{d x} (\frac{9}{( 2 x + 9 ) ^{2}})

\frac{d}{d x} (- 4 sin (4 x))

y = x - 1, y = - x + 3, x

\int x (x^{3} + 5 x) d x

\frac{d ^{2}}{d x ^{2}} (\frac{- x ^{2}}{x ^{2} - 2 x + 8})

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