Solve ∫ x^2left(7x^2-8right)^3 wrt x

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

Evaluate

$$\frac{343x^{9}}{9}-168x^{7}+\frac{1344x^{5}}{5}-\frac{512x^{3}}{3}+С$$

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}.

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

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

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

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

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

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

$$\int 343x^{8}-1176x^{6}+1344x^{4}-512x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 343x^{8}\mathrm{d}x+\int -1176x^{6}\mathrm{d}x+\int 1344x^{4}\mathrm{d}x+\int -512x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$343\int x^{8}\mathrm{d}x-1176\int x^{6}\mathrm{d}x+1344\int x^{4}\mathrm{d}x-512\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{8}\mathrm{d}x with \frac{x^{9}}{9}. Multiply 343 times \frac{x^{9}}{9}.

$$\frac{343x^{9}}{9}-1176\int x^{6}\mathrm{d}x+1344\int x^{4}\mathrm{d}x-512\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{6}\mathrm{d}x with \frac{x^{7}}{7}. Multiply -1176 times \frac{x^{7}}{7}.

$$\frac{343x^{9}}{9}-168x^{7}+1344\int x^{4}\mathrm{d}x-512\int x^{2}\mathrm{d}x$$

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 1344 times \frac{x^{5}}{5}.

$$\frac{343x^{9}}{9}-168x^{7}+\frac{1344x^{5}}{5}-512\int x^{2}\mathrm{d}x$$

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 -512 times \frac{x^{3}}{3}.

$$\frac{343x^{9}}{9}-168x^{7}+\frac{1344x^{5}}{5}-\frac{512x^{3}}{3}$$

Simplify.

$$-\frac{512x^{3}}{3}+\frac{1344x^{5}}{5}-168x^{7}+\frac{343x^{9}}{9}$$

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.

$$-\frac{512x^{3}}{3}+\frac{1344x^{5}}{5}-168x^{7}+\frac{343x^{9}}{9}+С$$

Differentiate w.r.t. x

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