Solve ∫ 12(y^3+4y+1)(y^2+4) wrt y

$$\int 12 ( y ^ { 3 } + 4 y + 1 ) ( y ^ { 2 } + 4 ) d y$$

Evaluate

$$2y^{6}+24y^{4}+4y^{3}+96y^{2}+48y+С$$

Solution Steps

Use the distributive property to multiply 12 by y^{3}+4y+1.

$$\int \left(12y^{3}+48y+12\right)\left(y^{2}+4\right)\mathrm{d}y$$

Use the distributive property to multiply 12y^{3}+48y+12 by y^{2}+4 and combine like terms.

$$\int 12y^{5}+96y^{3}+192y+12y^{2}+48\mathrm{d}y$$

Integrate the sum term by term.

$$\int 12y^{5}\mathrm{d}y+\int 96y^{3}\mathrm{d}y+\int 192y\mathrm{d}y+\int 12y^{2}\mathrm{d}y+\int 48\mathrm{d}y$$

Factor out the constant in each of the terms.

$$12\int y^{5}\mathrm{d}y+96\int y^{3}\mathrm{d}y+192\int y\mathrm{d}y+12\int y^{2}\mathrm{d}y+\int 48\mathrm{d}y$$

Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y^{5}\mathrm{d}y with \frac{y^{6}}{6}. Multiply 12 times \frac{y^{6}}{6}.

$$2y^{6}+96\int y^{3}\mathrm{d}y+192\int y\mathrm{d}y+12\int y^{2}\mathrm{d}y+\int 48\mathrm{d}y$$

Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y^{3}\mathrm{d}y with \frac{y^{4}}{4}. Multiply 96 times \frac{y^{4}}{4}.

$$2y^{6}+24y^{4}+192\int y\mathrm{d}y+12\int y^{2}\mathrm{d}y+\int 48\mathrm{d}y$$

Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y\mathrm{d}y with \frac{y^{2}}{2}. Multiply 192 times \frac{y^{2}}{2}.

$$2y^{6}+24y^{4}+96y^{2}+12\int y^{2}\mathrm{d}y+\int 48\mathrm{d}y$$

Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y^{2}\mathrm{d}y with \frac{y^{3}}{3}. Multiply 12 times \frac{y^{3}}{3}.

$$2y^{6}+24y^{4}+96y^{2}+4y^{3}+\int 48\mathrm{d}y$$

Find the integral of 48 using the table of common integrals rule \int a\mathrm{d}y=ay.

$$2y^{6}+24y^{4}+96y^{2}+4y^{3}+48y$$

If F\left(y\right) is an antiderivative of f\left(y\right), then the set of all antiderivatives of f\left(y\right) is given by F\left(y\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$2y^{6}+24y^{4}+96y^{2}+4y^{3}+48y+С$$

Differentiate w.r.t. y

$$12\left(y^{2}+4\right)\left(y^{3}+4y+1\right)$$