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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Add a constant to the solution

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

\frac{d}{d x} ((- 5 x)^{30})

\int_{- 4}^{5} 8 - y - \frac{( y - 2 ) ^{2}}{3} d y

\frac{d}{d x} (C x + C - C^{2})

x^{3} y^{^{'''}} - 2 x^{2} y^{^{''}} + 13 x y^{^{'}} - 13 y = 0

y = \frac{3}{x}, y = \frac{3}{x ^{2}}, x = 5

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