What is the integral of ((x+3))/((x^2+4)) ?

The integral of ((x+3))/((x^2+4)) is 1/2 ln|x^2+4|+3/2 arctan(x/2)+C

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integral of ((x+3))/((x^2+4))

integral

\frac{( x + 3 )}{( x ^{2} + 4 )}

\frac{1}{2} ln x^{2} + 4 + \frac{3}{2} arctan (\frac{x}{2}) + C

Solution steps

\int \frac{x + 3}{x ^{2} + 4} d x

Expand

\frac{x + 3}{x ^{2} + 4} : \frac{x}{x ^{2} + 4} + \frac{3}{x ^{2} + 4}

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}{x ^{2} + 4} d x + \int \frac{3}{x ^{2} + 4} d x

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

\int \frac{3}{x ^{2} + 4} d x = \frac{3}{2} arctan (\frac{x}{2})

= \frac{1}{2} ln x^{2} + 4 + \frac{3}{2} arctan (\frac{x}{2})

Add a constant to the solution

= \frac{1}{2} ln x^{2} + 4 + \frac{3}{2} arctan (\frac{x}{2}) + C

\frac{1}{( 2 x ^{3} )}

sec^{2} (2 x - 1)

y = [3 x + 2] [2 x - 5]

\frac{1}{( k + e ^{- t} )}

e^{\frac{y ^{2}}{2}}

What is the integral of ((x+3))/((x^2+4)) ?
The integral of ((x+3))/((x^2+4)) is 1/2 ln|x^2+4|+3/2 arctan(x/2)+C