What is the integral of (4x+3)/(x^2+9) ?

The integral of (4x+3)/(x^2+9) is 2ln|x^2+9|+arctan(x/3)+C

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integral of (4x+3)/(x^2+9)

\int \frac{4 x + 3}{x ^{2} + 9} d x

2 ln x^{2} + 9 + arctan (\frac{x}{3}) + C

Solution steps

\int \frac{4 x + 3}{x ^{2} + 9} d x

Expand

\frac{4 x + 3}{x ^{2} + 9} : \frac{4 x}{x ^{2} + 9} + \frac{3}{x ^{2} + 9}

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{4 x}{x ^{2} + 9} d x + \int \frac{3}{x ^{2} + 9} d x

\int \frac{4 x}{x ^{2} + 9} d x = 2 ln x^{2} + 9

\int \frac{3}{x ^{2} + 9} d x = arctan (\frac{x}{3})

= 2 ln x^{2} + 9 + arctan (\frac{x}{3})

Add a constant to the solution

= 2 ln x^{2} + 9 + arctan (\frac{x}{3}) + C

\frac{d ^{2}}{d x ^{2}} (\frac{7}{8 x ^{5}})

\int (4 x^{2} - 2) \cdot x d x

\int_{2}^{4} (\frac{5}{x} - \frac{10}{x ^{2}}) d x

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

y^{^{'}} + a \cdot y = b

What is the integral of (4x+3)/(x^2+9) ?
The integral of (4x+3)/(x^2+9) is 2ln|x^2+9|+arctan(x/3)+C