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

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

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

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

1 + u^{2} + C

Solution steps

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

Apply u-substitution

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

= \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

: 2 v^{\frac{1}{2}}

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

Substitute back

v = 1 + u^{2}

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

Simplify

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

= 1 + u^{2}

Add a constant to the solution

= 1 + u^{2} + C

f (x) = x^{3} - 12 x

\frac{d}{d x} (2 cos (2 x))

\frac{d}{d x} (\frac{4}{x} + \frac{5}{x ^{2}})

\frac{\partial}{\partial x} (1 + x ln (x y - 7))

\frac{d x}{d t} = 30 - \frac{3 x}{100}

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