Solve ∫ x(6x^2-1)^3 wrt x

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

Evaluate

$$27x^{8}-18x^{6}+\frac{9x^{4}}{2}-\frac{x^{2}}{2}+С$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(6x^{2}-1\right)^{3}.

$$\int x\left(216\left(x^{2}\right)^{3}-108\left(x^{2}\right)^{2}+18x^{2}-1\right)\mathrm{d}x$$

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

$$\int x\left(216x^{6}-108\left(x^{2}\right)^{2}+18x^{2}-1\right)\mathrm{d}x$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\int x\left(216x^{6}-108x^{4}+18x^{2}-1\right)\mathrm{d}x$$

Use the distributive property to multiply x by 216x^{6}-108x^{4}+18x^{2}-1.

$$\int 216x^{7}-108x^{5}+18x^{3}-x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 216x^{7}\mathrm{d}x+\int -108x^{5}\mathrm{d}x+\int 18x^{3}\mathrm{d}x+\int -x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$216\int x^{7}\mathrm{d}x-108\int x^{5}\mathrm{d}x+18\int x^{3}\mathrm{d}x-\int x\mathrm{d}x$$

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 216 times \frac{x^{8}}{8}.

$$27x^{8}-108\int x^{5}\mathrm{d}x+18\int x^{3}\mathrm{d}x-\int x\mathrm{d}x$$

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 -108 times \frac{x^{6}}{6}.

$$27x^{8}-18x^{6}+18\int x^{3}\mathrm{d}x-\int x\mathrm{d}x$$

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}.

$$27x^{8}-18x^{6}+\frac{9x^{4}}{2}-\int x\mathrm{d}x$$

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 -1 times \frac{x^{2}}{2}.

$$27x^{8}-18x^{6}+\frac{9x^{4}}{2}-\frac{x^{2}}{2}$$

Simplify.

$$-\frac{x^{2}}{2}+\frac{9x^{4}}{2}-18x^{6}+27x^{8}$$

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.

$$-\frac{x^{2}}{2}+\frac{9x^{4}}{2}-18x^{6}+27x^{8}+С$$

Differentiate w.r.t. x

$$x\left(6x^{2}-1\right)^{3}$$