Solve ∫ 3x^2(x^3+8)^3 wrt x=?

$$\int 3 x ^ { 2 } ( x ^ { 3 } + 8 ) ^ { 3 } d x = ?$$

Evaluate

$$\frac{x^{12}}{4}+8x^{9}+96x^{6}+512x^{3}+С$$

Solution Steps

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

$$\int 3x^{2}\left(\left(x^{3}\right)^{3}+24\left(x^{3}\right)^{2}+192x^{3}+512\right)\mathrm{d}x$$

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

$$\int 3x^{2}\left(x^{9}+24\left(x^{3}\right)^{2}+192x^{3}+512\right)\mathrm{d}x$$

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

$$\int 3x^{2}\left(x^{9}+24x^{6}+192x^{3}+512\right)\mathrm{d}x$$

Use the distributive property to multiply 3x^{2} by x^{9}+24x^{6}+192x^{3}+512.

$$\int 3x^{11}+72x^{8}+576x^{5}+1536x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 3x^{11}\mathrm{d}x+\int 72x^{8}\mathrm{d}x+\int 576x^{5}\mathrm{d}x+\int 1536x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$3\int x^{11}\mathrm{d}x+72\int x^{8}\mathrm{d}x+576\int x^{5}\mathrm{d}x+1536\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{11}\mathrm{d}x with \frac{x^{12}}{12}. Multiply 3 times \frac{x^{12}}{12}.

$$\frac{x^{12}}{4}+72\int x^{8}\mathrm{d}x+576\int x^{5}\mathrm{d}x+1536\int x^{2}\mathrm{d}x$$

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 72 times \frac{x^{9}}{9}.

$$\frac{x^{12}}{4}+8x^{9}+576\int x^{5}\mathrm{d}x+1536\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{5}\mathrm{d}x with \frac{x^{6}}{6}. Multiply 576 times \frac{x^{6}}{6}.

$$\frac{x^{12}}{4}+8x^{9}+96x^{6}+1536\int x^{2}\mathrm{d}x$$

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 1536 times \frac{x^{3}}{3}.

$$\frac{x^{12}}{4}+8x^{9}+96x^{6}+512x^{3}$$

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.

$$512x^{3}+96x^{6}+8x^{9}+\frac{x^{12}}{4}+С$$

Differentiate w.r.t. x

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