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

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

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

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

u^{2} + 1 + C

Solution steps

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

Apply u-substitution

= \int \frac{1}{2 v} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1}{v} d v

Apply the Power Rule

= \frac{1}{2} \cdot 2 v^{\frac{1}{2}}

Substitute back

v = u^{2} + 1

= \frac{1}{2} \cdot 2 (u^{2} + 1)^{\frac{1}{2}}

Simplify

\frac{1}{2} \cdot 2 (u^{2} + 1)^{\frac{1}{2}} : u^{2} + 1

= u^{2} + 1

Add a constant to the solution

= u^{2} + 1 + C

x \to 0 lim (\frac{ln ( \frac{1 + x}{1 - x} ) - 2 x}{x - sin ( x )})

\int x x^{4} + 1 d x

f (x) = 6 x - x^{2}, at x = 0

f (t) = 80 t + t^{2} - \frac{1}{48} t^{3}

- 4 x + x^{2} + 10, 12 - 5 x, 4 x - 6

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