Solve ∫ 9x(5x^2+10)^3 wrt x=?

$$\int 9 x ( 5 x ^ { 2 } + 10 ) ^ { 3 } d x = ?$$

Evaluate

$$\frac{1125x^{8}}{8}+1125x^{6}+3375x^{4}+4500x^{2}+С$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(5x^{2}+10\right)^{3}.

$$\int 9x\left(125\left(x^{2}\right)^{3}+750\left(x^{2}\right)^{2}+1500x^{2}+1000\right)\mathrm{d}x$$

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

$$\int 9x\left(125x^{6}+750\left(x^{2}\right)^{2}+1500x^{2}+1000\right)\mathrm{d}x$$

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

$$\int 9x\left(125x^{6}+750x^{4}+1500x^{2}+1000\right)\mathrm{d}x$$

Use the distributive property to multiply 9x by 125x^{6}+750x^{4}+1500x^{2}+1000.

$$\int 1125x^{7}+6750x^{5}+13500x^{3}+9000x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 1125x^{7}\mathrm{d}x+\int 6750x^{5}\mathrm{d}x+\int 13500x^{3}\mathrm{d}x+\int 9000x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$1125\int x^{7}\mathrm{d}x+6750\int x^{5}\mathrm{d}x+13500\int x^{3}\mathrm{d}x+9000\int x\mathrm{d}x$$

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 1125 times \frac{x^{8}}{8}.

$$\frac{1125x^{8}}{8}+6750\int x^{5}\mathrm{d}x+13500\int x^{3}\mathrm{d}x+9000\int x\mathrm{d}x$$

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 6750 times \frac{x^{6}}{6}.

$$\frac{1125x^{8}}{8}+1125x^{6}+13500\int x^{3}\mathrm{d}x+9000\int x\mathrm{d}x$$

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 13500 times \frac{x^{4}}{4}.

$$\frac{1125x^{8}}{8}+1125x^{6}+3375x^{4}+9000\int x\mathrm{d}x$$

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 9000 times \frac{x^{2}}{2}.

$$\frac{1125x^{8}}{8}+1125x^{6}+3375x^{4}+4500x^{2}$$

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.

$$\frac{1125x^{8}}{8}+1125x^{6}+3375x^{4}+4500x^{2}+С$$

Differentiate w.r.t. x

$$1125x\left(x^{2}+2\right)^{3}$$