What is the integral of (5+9x)/(1+x^2) ?

The integral of (5+9x)/(1+x^2) is 5arctan(x)+9/2 ln|1+x^2|+C

Popular >

integral of (5+9x)/(1+x^2)

\int \frac{5 + 9 x}{1 + x ^{2}} d x

5 arctan (x) + \frac{9}{2} ln 1 + x^{2} + C

Solution steps

\int \frac{5 + 9 x}{1 + x ^{2}} d x

Expand

\frac{5 + 9 x}{1 + x ^{2}} : \frac{5}{1 + x ^{2}} + \frac{9 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{5}{1 + x ^{2}} d x + \int \frac{9 x}{1 + x ^{2}} d x

\int \frac{5}{1 + x ^{2}} d x = 5 arctan (x)

\int \frac{9 x}{1 + x ^{2}} d x = \frac{9}{2} ln 1 + x^{2}

= 5 arctan (x) + \frac{9}{2} ln 1 + x^{2}

Add a constant to the solution

= 5 arctan (x) + \frac{9}{2} ln 1 + x^{2} + C

\int \frac{1}{9 + 25 x ^{2}} d x

\int (\frac{5 x + 3}{x ^{2} + 2 x - 3}) d x

\int \frac{( x + 8 ) ^{4}}{2 x} d x

\int \frac{e ^{x}}{( e ^{x} + 1 ) ^{17}} d x

\int \frac{x ^{2}}{x ^{6} + 4} d x

What is the integral of (5+9x)/(1+x^2) ?
The integral of (5+9x)/(1+x^2) is 5arctan(x)+9/2 ln|1+x^2|+C