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

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

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

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

\frac{1}{8} ln 2 + 4 x^{2} + \frac{1}{2 2} arctan (2 x) + C

Solution steps

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

Expand

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

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

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

= \frac{1}{8} ln 2 + 4 x^{2} + \frac{1}{2 2} arctan (2 x)

Add a constant to the solution

= \frac{1}{8} ln 2 + 4 x^{2} + \frac{1}{2 2} arctan (2 x) + C

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

\int \frac{( sin ( w ) )}{w} d w

\frac{( s - 2 )}{( s - 2 ) ^{2} - 3}

\int \frac{( e ^{x} )}{1 + e ^{x}} d x

y = 4 t t + 7

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