What is the integral of (u^2-1)/(u^2+1) ?

The integral of (u^2-1)/(u^2+1) is -2arctan(u)+u+C

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integral of (u^2-1)/(u^2+1)

\int \frac{u ^{2} - 1}{u ^{2} + 1} d u

- 2 arctan (u) + u + C

Solution steps

\int \frac{u ^{2} - 1}{u ^{2} + 1} d u

\frac{u ^{2} - 1}{u ^{2} + 1} = - \frac{2}{u ^{2} + 1} + 1

= \int - \frac{2}{u ^{2} + 1} + 1 d u

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{2}{u ^{2} + 1} d u + \int 1 d u

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

\int 1 d u = u

= - 2 arctan (u) + u

Add a constant to the solution

= - 2 arctan (u) + u + C

f (x) = 5 e^{x}, at x = 3

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

\frac{d}{d x} (\frac{x ^{3}}{( x + 2 ) ^{2}})

\int 36 (4 x + 4)^{(2)} - 2 (3 x - 1) d x

\int cos^{- 1} (x) d x

What is the integral of (u^2-1)/(u^2+1) ?
The integral of (u^2-1)/(u^2+1) is -2arctan(u)+u+C