What is the integral of (3s)/(sqrt(1+s^2)) ?

The integral of (3s)/(sqrt(1+s^2)) is 3sqrt(1+s^2)+C

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integral of (3s)/(sqrt(1+s^2))

\int \frac{3 s}{1 + s ^{2}} d s

3 1 + s^{2} + C

Solution steps

\int \frac{3 s}{1 + s ^{2}} d s

Take the constant out:

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

= 3 \cdot \int \frac{s}{1 + s ^{2}} d s

Apply u-substitution

= 3 \cdot \int \frac{1}{2 u} d u

Take the constant out:

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

= 3 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

Apply the Power Rule

= 3 \cdot \frac{1}{2} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 1 + s^{2}

= 3 \cdot \frac{1}{2} \cdot 2 (1 + s^{2})^{\frac{1}{2}}

Simplify

3 \cdot \frac{1}{2} \cdot 2 (1 + s^{2})^{\frac{1}{2}} : 3 1 + s^{2}

= 3 1 + s^{2}

Add a constant to the solution

= 3 1 + s^{2} + C

What is the integral of (3s)/(sqrt(1+s^2)) ?
The integral of (3s)/(sqrt(1+s^2)) is 3sqrt(1+s^2)+C