What is the integral of 8(y^4+4y^2+1)^2(y^3+2y) ?

The integral of 8(y^4+4y^2+1)^2(y^3+2y) is 2/3 (y^4+4y^2+1)^3+C

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integral of 8(y^4+4y^2+1)^2(y^3+2y)

\int 8 (y^{4} + 4 y^{2} + 1)^{2} (y^{3} + 2 y) d y

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

Solution steps

\int 8 (y^{4} + 4 y^{2} + 1)^{2} (y^{3} + 2 y) d y

Take the constant out:

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

= 8 \cdot \int (y^{4} + 4 y^{2} + 1)^{2} (y^{3} + 2 y) d y

Apply u-substitution

= 8 \cdot \int \frac{u ^{2}}{4} d u

Take the constant out:

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

= 8 \cdot \frac{1}{4} \cdot \int u^{2} d u

Apply the Power Rule

= 8 \cdot \frac{1}{4} \cdot \frac{u ^{3}}{3}

Substitute back

u = y^{4} + 4 y^{2} + 1

= 8 \cdot \frac{1}{4} \cdot \frac{( y ^{4} + 4 y ^{2} + 1 ) ^{3}}{3}

Simplify

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

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

Add a constant to the solution

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

What is the integral of 8(y^4+4y^2+1)^2(y^3+2y) ?
The integral of 8(y^4+4y^2+1)^2(y^3+2y) is 2/3 (y^4+4y^2+1)^3+C