What is the integral of ((8+5x))/(1+x^2) ?

The integral of ((8+5x))/(1+x^2) is 8arctan(x)+5/2 ln|1+x^2|+C

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integral of ((8+5x))/(1+x^2)

\int \frac{( 8 + 5 x )}{1 + x ^{2}} d x

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

Solution steps

\int \frac{8 + 5 x}{1 + x ^{2}} d x

Expand

\frac{8 + 5 x}{1 + x ^{2}} : \frac{8}{1 + x ^{2}} + \frac{5 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{8}{1 + x ^{2}} d x + \int \frac{5 x}{1 + x ^{2}} d x

\int \frac{8}{1 + x ^{2}} d x = 8 arctan (x)

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

= 8 arctan (x) + \frac{5}{2} ln 1 + x^{2}

Add a constant to the solution

= 8 arctan (x) + \frac{5}{2} ln 1 + x^{2} + C

\frac{d}{d x} ([x + (x + sin^{2} (x))^{6}]^{3})

\frac{\partial}{\partial y} (x e^{x y + 1})

\int_{- 1}^{x^{3}} 2 + t^{2} d t

(x^{2} + 1) y^{^{'}} + 6 x (y - 1) = 0, y (0) = 9

(x - 1) ln (x), 1

What is the integral of ((8+5x))/(1+x^2) ?
The integral of ((8+5x))/(1+x^2) is 8arctan(x)+5/2 ln|1+x^2|+C