integral of 12(y^4+4y^2+8)^2(y^3+2y)

Let's integrate:

\int 12 \cdot {(y^{4} + 4 y^{2} + 8)}^{2} \cdot (y^{3} + 2 y) d y

Let

u = y^{4} + 4 y^{2} + 8

. Then

du = (4 y^{3} + 8 y) \cdot dy

, so

12 \cdot (y^{3} + 2 y) \cdot dy = 3 du

\int 12 \cdot {(y^{4} + 4 y^{2} + 8)}^{2} \cdot (y^{3} + 2 y) d y = 3 \cdot \int u^{2} d u

An antiderivative of

u^{2}

\frac{u^{3}}{3}

Substitute

u

y^{4} + 4 y^{2} + 8

back into the antiderivative

ⅰ

\int 12 \cdot {(y^{4} + 4 y^{2} + 8)}^{2} \cdot (y^{3} + 2 y) d y

{(y^{4} + 4 y^{2} + 8)}^{3}

{(y^{4} + 4 y^{2} + 8)}^{3}

▶Indefinite integrals are defined up to an additive constant C, so this yields:

{(y^{4} + 4 y^{2} + 8)}^{3} + C