What is the integral of (5-3x)/(x^2+4x+5) ?

The integral of (5-3x)/(x^2+4x+5) is -3/2 ln|x^2+4x+5|+11arctan(x+2)+C

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integral of (5-3x)/(x^2+4x+5)

\int \frac{5 - 3 x}{x ^{2} + 4 x + 5} d x

- \frac{3}{2} ln x^{2} + 4 x + 5 + 11 arctan (x + 2) + C

Solution steps

\int \frac{5 - 3 x}{x ^{2} + 4 x + 5} d x

Expand

\frac{5 - 3 x}{x ^{2} + 4 x + 5} : \frac{5}{x ^{2} + 4 x + 5} - \frac{3 x}{x ^{2} + 4 x + 5}

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}{x ^{2} + 4 x + 5} d x - \int \frac{3 x}{x ^{2} + 4 x + 5} d x

\int \frac{5}{x ^{2} + 4 x + 5} d x = 5 arctan (x + 2)

\int \frac{3 x}{x ^{2} + 4 x + 5} d x = 3 (\frac{1}{2} ln x^{2} + 4 x + 5 - 2 arctan (x + 2))

= 5 arctan (x + 2) - 3 (\frac{1}{2} ln x^{2} + 4 x + 5 - 2 arctan (x + 2))

Simplify

5 arctan (x + 2) - 3 (\frac{1}{2} ln x^{2} + 4 x + 5 - 2 arctan (x + 2)) : - \frac{3}{2} ln x^{2} + 4 x + 5 + 11 arctan (x + 2)

= - \frac{3}{2} ln x^{2} + 4 x + 5 + 11 arctan (x + 2)

Add a constant to the solution

= - \frac{3}{2} ln x^{2} + 4 x + 5 + 11 arctan (x + 2) + C

\int \frac{5}{3 - 2 x} d x

\int (x^{2} + 3 x + 2) ln (x) d x

\int \frac{2 x ^{2} - 4}{x ^{3}} d x

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

\int 4 1 - 5 x^{2} d x

What is the integral of (5-3x)/(x^2+4x+5) ?
The integral of (5-3x)/(x^2+4x+5) is -3/2 ln|x^2+4x+5|+11arctan(x+2)+C