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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

y = 1 - x^{2} arcsin (x)

y^{^{'}} = (x^{2})

y^{^{'}} = \frac{( 5 x sec ( \frac{y}{x} ) + y )}{x}

\frac{d}{d x} (\frac{y}{1 - 2 x})

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

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