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

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