What is the integral of (8x+1)/(x^2+1) ?

The integral of (8x+1)/(x^2+1) is 4ln|x^2+1|+arctan(x)+C

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integral of (8x+1)/(x^2+1)

\int \frac{8 x + 1}{x ^{2} + 1} d x

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

Solution steps

\int \frac{8 x + 1}{x ^{2} + 1} d x

Expand

\frac{8 x + 1}{x ^{2} + 1} : \frac{8 x}{x ^{2} + 1} + \frac{1}{x ^{2} + 1}

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{8 x}{x ^{2} + 1} d x + \int \frac{1}{x ^{2} + 1} d x

\int \frac{8 x}{x ^{2} + 1} d x = 4 ln x^{2} + 1

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

= 4 ln x^{2} + 1 + arctan (x)

Add a constant to the solution

= 4 ln x^{2} + 1 + arctan (x) + C

f (x) = \frac{8}{x}, at x = 5

y^{^{''}} + y = 9 sin (x)

40 ln (3 x + 1) - \frac{x}{3}

y^{^{''}} - 7 y^{^{'}} + 7 y = 0

\int 3 x - 5 d x

What is the integral of (8x+1)/(x^2+1) ?
The integral of (8x+1)/(x^2+1) is 4ln|x^2+1|+arctan(x)+C