What is the integral of (x+5)/(x^2+16) ?

The integral of (x+5)/(x^2+16) is 1/2 ln|x^2+16|+5/4 arctan(x/4)+C

integral of (x+5)/(x^2+16)

\int \frac{x + 5}{x ^{2} + 16} d x

\frac{1}{2} ln x^{2} + 16 + \frac{5}{4} arctan (\frac{x}{4}) + C

Solution steps

\int \frac{x + 5}{x ^{2} + 16} d x

Expand

\frac{x + 5}{x ^{2} + 16} : \frac{x}{x ^{2} + 16} + \frac{5}{x ^{2} + 16}

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{x}{x ^{2} + 16} d x + \int \frac{5}{x ^{2} + 16} d x

\int \frac{x}{x ^{2} + 16} d x = \frac{1}{2} ln x^{2} + 16

\int \frac{5}{x ^{2} + 16} d x = \frac{5}{4} arctan (\frac{x}{4})

= \frac{1}{2} ln x^{2} + 16 + \frac{5}{4} arctan (\frac{x}{4})

Add a constant to the solution

= \frac{1}{2} ln x^{2} + 16 + \frac{5}{4} arctan (\frac{x}{4}) + C

d er i v a t i v e f (x) = \frac{( x ^{2} + 3 )}{x}

s l o p e in t erce pt (- 1, 1), (- 2, - 1)

d er i v a t i v e y = x + 5

x (\frac{d y}{d x}) - y = x^{2} sin (x)

y d x + (x - 1) d y = 0, y (2) = 9

What is the integral of (x+5)/(x^2+16) ?
The integral of (x+5)/(x^2+16) is 1/2 ln|x^2+16|+5/4 arctan(x/4)+C