What is the integral of (5+3x)/(1+x^2) ?

The integral of (5+3x)/(1+x^2) is 5arctan(x)+3/2 ln|1+x^2|+C

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integral of (5+3x)/(1+x^2)

\int \frac{5 + 3 x}{1 + x ^{2}} d x

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

Solution steps

\int \frac{5 + 3 x}{1 + x ^{2}} d x

Expand

\frac{5 + 3 x}{1 + x ^{2}} : \frac{5}{1 + x ^{2}} + \frac{3 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{3 x}{1 + x ^{2}} d x

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

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

= 5 arctan (x) + \frac{3}{2} ln 1 + x^{2}

Add a constant to the solution

= 5 arctan (x) + \frac{3}{2} ln 1 + x^{2} + C

\int 15 x + 1 d x

e^{x} y \frac{d y}{d x} = 1

y = \frac{2}{x}, (- 2, - 1)

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

x \to 1 lim (5 x^{5} - 8 x^{3} + 8)

What is the integral of (5+3x)/(1+x^2) ?
The integral of (5+3x)/(1+x^2) is 5arctan(x)+3/2 ln|1+x^2|+C