Solve ∫ left(7x^2-8right)^3x wrt x

$$\int{ { \left(7 { x }^{ 2 } -8 \right) }^{ 3 } x }d x$$

Evaluate

$$\frac{343x^{8}}{8}-196x^{6}+336x^{4}-256x^{2}+С$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(7x^{2}-8\right)^{3}.

$$\int \left(343\left(x^{2}\right)^{3}-1176\left(x^{2}\right)^{2}+1344x^{2}-512\right)x\mathrm{d}x$$

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

$$\int \left(343x^{6}-1176\left(x^{2}\right)^{2}+1344x^{2}-512\right)x\mathrm{d}x$$

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

$$\int \left(343x^{6}-1176x^{4}+1344x^{2}-512\right)x\mathrm{d}x$$

Use the distributive property to multiply 343x^{6}-1176x^{4}+1344x^{2}-512 by x.

$$\int 343x^{7}-1176x^{5}+1344x^{3}-512x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 343x^{7}\mathrm{d}x+\int -1176x^{5}\mathrm{d}x+\int 1344x^{3}\mathrm{d}x+\int -512x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$343\int x^{7}\mathrm{d}x-1176\int x^{5}\mathrm{d}x+1344\int x^{3}\mathrm{d}x-512\int x\mathrm{d}x$$

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

$$\frac{343x^{8}}{8}-1176\int x^{5}\mathrm{d}x+1344\int x^{3}\mathrm{d}x-512\int x\mathrm{d}x$$

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 -1176 times \frac{x^{6}}{6}.

$$\frac{343x^{8}}{8}-196x^{6}+1344\int x^{3}\mathrm{d}x-512\int x\mathrm{d}x$$

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 1344 times \frac{x^{4}}{4}.

$$\frac{343x^{8}}{8}-196x^{6}+336x^{4}-512\int x\mathrm{d}x$$

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 -512 times \frac{x^{2}}{2}.

$$\frac{343x^{8}}{8}-196x^{6}+336x^{4}-256x^{2}$$

Simplify.

$$-256x^{2}+336x^{4}-196x^{6}+\frac{343x^{8}}{8}$$

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.

$$-256x^{2}+336x^{4}-196x^{6}+\frac{343x^{8}}{8}+С$$

Differentiate w.r.t. x

$$x\left(7x^{2}-8\right)^{3}$$