Solve ∫ (3x^2+6x)(x^3+3x^2+1) wrt x

$$\int ( 3 x ^ { 2 } + 6 x ) ( x ^ { 3 } + 3 x ^ { 2 } + 1 ) d x$$

Evaluate

$$\frac{x^{6}}{2}+3x^{5}+\frac{9x^{4}}{2}+x^{3}+3x^{2}+С$$

Solution Steps

Use the distributive property to multiply 3x^{2}+6x by x^{3}+3x^{2}+1 and combine like terms.

$$\int 3x^{5}+15x^{4}+3x^{2}+18x^{3}+6x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 3x^{5}\mathrm{d}x+\int 15x^{4}\mathrm{d}x+\int 3x^{2}\mathrm{d}x+\int 18x^{3}\mathrm{d}x+\int 6x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$3\int x^{5}\mathrm{d}x+15\int x^{4}\mathrm{d}x+3\int x^{2}\mathrm{d}x+18\int x^{3}\mathrm{d}x+6\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{5}\mathrm{d}x with \frac{x^{6}}{6}. Multiply 3 times \frac{x^{6}}{6}.

$$\frac{x^{6}}{2}+15\int x^{4}\mathrm{d}x+3\int x^{2}\mathrm{d}x+18\int x^{3}\mathrm{d}x+6\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{4}\mathrm{d}x with \frac{x^{5}}{5}. Multiply 15 times \frac{x^{5}}{5}.

$$\frac{x^{6}}{2}+3x^{5}+3\int x^{2}\mathrm{d}x+18\int x^{3}\mathrm{d}x+6\int x\mathrm{d}x$$

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

$$\frac{x^{6}}{2}+3x^{5}+x^{3}+18\int x^{3}\mathrm{d}x+6\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. Multiply 18 times \frac{x^{4}}{4}.

$$\frac{x^{6}}{2}+3x^{5}+x^{3}+\frac{9x^{4}}{2}+6\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply 6 times \frac{x^{2}}{2}.

$$\frac{x^{6}}{2}+3x^{5}+x^{3}+\frac{9x^{4}}{2}+3x^{2}$$

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

$$\frac{x^{6}}{2}+3x^{5}+x^{3}+\frac{9x^{4}}{2}+3x^{2}+С$$

Differentiate w.r.t. x

$$3x\left(x+2\right)\left(x^{3}+3x^{2}+1\right)$$