Solve ∫ 3x^2(x^3+6)^3 wrt x=?

$$\int 3 x ^ { 2 } ( x ^ { 3 } + 6 ) ^ { 3 } d x = ?$$

Evaluate

$$\frac{x^{12}}{4}+6x^{9}+54x^{6}+216x^{3}+С$$

Solution Steps

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

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

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

$$\int 3x^{2}\left(x^{9}+18\left(x^{3}\right)^{2}+108x^{3}+216\right)\mathrm{d}x$$

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

$$\int 3x^{2}\left(x^{9}+18x^{6}+108x^{3}+216\right)\mathrm{d}x$$

Use the distributive property to multiply 3x^{2} by x^{9}+18x^{6}+108x^{3}+216.

$$\int 3x^{11}+54x^{8}+324x^{5}+648x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 3x^{11}\mathrm{d}x+\int 54x^{8}\mathrm{d}x+\int 324x^{5}\mathrm{d}x+\int 648x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$3\int x^{11}\mathrm{d}x+54\int x^{8}\mathrm{d}x+324\int x^{5}\mathrm{d}x+648\int x^{2}\mathrm{d}x$$

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}.

$$\frac{x^{12}}{4}+54\int x^{8}\mathrm{d}x+324\int x^{5}\mathrm{d}x+648\int x^{2}\mathrm{d}x$$

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

$$\frac{x^{12}}{4}+6x^{9}+324\int x^{5}\mathrm{d}x+648\int x^{2}\mathrm{d}x$$

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

$$\frac{x^{12}}{4}+6x^{9}+54x^{6}+648\int x^{2}\mathrm{d}x$$

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

$$\frac{x^{12}}{4}+6x^{9}+54x^{6}+216x^{3}$$

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.

$$216x^{3}+54x^{6}+6x^{9}+\frac{x^{12}}{4}+С$$

Differentiate w.r.t. x

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