What is the integral of (5+4x)/(1+x^2) ?

The integral of (5+4x)/(1+x^2) is 5arctan(x)+2ln|1+x^2|+C

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integral of (5+4x)/(1+x^2)

\int \frac{5 + 4 x}{1 + x ^{2}} d x

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

Solution steps

\int \frac{5 + 4 x}{1 + x ^{2}} d x

Expand

\frac{5 + 4 x}{1 + x ^{2}} : \frac{5}{1 + x ^{2}} + \frac{4 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{4 x}{1 + x ^{2}} d x

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

\int \frac{4 x}{1 + x ^{2}} d x = 2 ln 1 + x^{2}

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

Add a constant to the solution

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

\int e^{- \frac{x}{8}} d x

x^{^{''}} + 2 x^{^{'}} + 2 = 0, x (0) = 0, x^{^{'}} (0) = 1

x \to 0 lim (\frac{1}{x ^{3} - 7 x})

f (x) = 2 x^{3} + 3 x^{2} - 5 x, - 4, 3

\frac{\partial}{\partial x} (x^{3} ln (yz))

What is the integral of (5+4x)/(1+x^2) ?
The integral of (5+4x)/(1+x^2) is 5arctan(x)+2ln|1+x^2|+C