What is the integral of (3+x)/(1+x^2) ?

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

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integral of (3+x)/(1+x^2)

\int \frac{3 + x}{1 + x ^{2}} d x

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

Solution steps

\int \frac{3 + x}{1 + x ^{2}} d x

Expand

\frac{3 + x}{1 + x ^{2}} : \frac{3}{1 + x ^{2}} + \frac{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{3}{1 + x ^{2}} d x + \int \frac{x}{1 + x ^{2}} d x

\int \frac{3}{1 + x ^{2}} d x = 3 arctan (x)

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

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

Add a constant to the solution

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

x^{2} + 10, 9 x, 0

\int \frac{( 9 - y ^{2} ) ^{2}}{2} - y (9 - y^{2}) d y

\int y + 4 d y

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

\int_{1}^{4} - 10 t ln (t) d t

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