Solve ∫ left(x^3-1right)^32x wrt x

$$\int{ { \left( { x }^{ 3 } -1 \right) }^{ 3 } 2x }d x$$

Evaluate

$$\frac{2x^{11}}{11}-\frac{3x^{8}}{4}+\frac{6x^{5}}{5}-x^{2}+С$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x^{3}-1\right)^{3}.

$$\int \left(\left(x^{3}\right)^{3}-3\left(x^{3}\right)^{2}+3x^{3}-1\right)\times 2x\mathrm{d}x$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$\int \left(x^{9}-3\left(x^{3}\right)^{2}+3x^{3}-1\right)\times 2x\mathrm{d}x$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\int \left(x^{9}-3x^{6}+3x^{3}-1\right)\times 2x\mathrm{d}x$$

Use the distributive property to multiply x^{9}-3x^{6}+3x^{3}-1 by 2.

$$\int \left(2x^{9}-6x^{6}+6x^{3}-2\right)x\mathrm{d}x$$

Use the distributive property to multiply 2x^{9}-6x^{6}+6x^{3}-2 by x.

$$\int 2x^{10}-6x^{7}+6x^{4}-2x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2x^{10}\mathrm{d}x+\int -6x^{7}\mathrm{d}x+\int 6x^{4}\mathrm{d}x+\int -2x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x^{10}\mathrm{d}x-6\int x^{7}\mathrm{d}x+6\int x^{4}\mathrm{d}x-2\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{10}\mathrm{d}x with \frac{x^{11}}{11}. Multiply 2 times \frac{x^{11}}{11}.

$$\frac{2x^{11}}{11}-6\int x^{7}\mathrm{d}x+6\int x^{4}\mathrm{d}x-2\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{7}\mathrm{d}x with \frac{x^{8}}{8}. Multiply -6 times \frac{x^{8}}{8}.

$$\frac{2x^{11}}{11}-\frac{3x^{8}}{4}+6\int x^{4}\mathrm{d}x-2\int x\mathrm{d}x$$

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 6 times \frac{x^{5}}{5}.

$$\frac{2x^{11}}{11}-\frac{3x^{8}}{4}+\frac{6x^{5}}{5}-2\int x\mathrm{d}x$$

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 -2 times \frac{x^{2}}{2}.

$$\frac{2x^{11}}{11}-\frac{3x^{8}}{4}+\frac{6x^{5}}{5}-x^{2}$$

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.

$$\frac{2x^{11}}{11}-\frac{3x^{8}}{4}+\frac{6x^{5}}{5}-x^{2}+С$$

Differentiate w.r.t. x

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