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

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

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

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

\frac{2}{3} s^{3} + 3 s^{2} + 1 + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{3 u} d u

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

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

u = s^{3} + 3 s^{2} + 1

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

Simplify

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

= \frac{2}{3} s^{3} + 3 s^{2} + 1

Add a constant to the solution

= \frac{2}{3} s^{3} + 3 s^{2} + 1 + C

9 sin (x)

\int \frac{cos ( 5 x )}{1 - sin ( 5 x )} d x

\int \frac{x ^{\frac{3}{2}}}{2} d x

\int \frac{x - 1}{2 x - x ^{2}} d x

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

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