What is the integral of (y^2+2y)/(sqrt(y^3+3y^2+4)) ?

The integral of (y^2+2y)/(sqrt(y^3+3y^2+4)) is 2/3 sqrt(y^3+3y^2+4)+C

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integral of (y^2+2y)/(sqrt(y^3+3y^2+4))

\int \frac{y ^{2} + 2 y}{y ^{3} + 3 y ^{2} + 4} d y

\frac{2}{3} y^{3} + 3 y^{2} + 4 + C

Solution steps

\int \frac{y ^{2} + 2 y}{y ^{3} + 3 y ^{2} + 4} d y

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 = y^{3} + 3 y^{2} + 4

= \frac{1}{3} \cdot 2 (y^{3} + 3 y^{2} + 4)^{\frac{1}{2}}

Simplify

\frac{1}{3} \cdot 2 (y^{3} + 3 y^{2} + 4)^{\frac{1}{2}} : \frac{2}{3} y^{3} + 3 y^{2} + 4

= \frac{2}{3} y^{3} + 3 y^{2} + 4

Add a constant to the solution

= \frac{2}{3} y^{3} + 3 y^{2} + 4 + C

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What is the integral of (y^2+2y)/(sqrt(y^3+3y^2+4)) ?
The integral of (y^2+2y)/(sqrt(y^3+3y^2+4)) is 2/3 sqrt(y^3+3y^2+4)+C