Solve ∫ (8x+2x^3)(4x^6+6x^7) wrt x=?

$$\int ( 8 x + 2 x ^ { 3 } ) ( 4 x ^ { 6 } + 6 x ^ { 7 } ) d x = ?$$

Evaluate

$$\frac{12x^{11}}{11}+\frac{4x^{10}}{5}+\frac{16x^{9}}{3}+4x^{8}+С$$

Solution Steps

Use the distributive property to multiply 8x+2x^{3} by 4x^{6}+6x^{7}.

$$\int 32x^{7}+48x^{8}+8x^{9}+12x^{10}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 32x^{7}\mathrm{d}x+\int 48x^{8}\mathrm{d}x+\int 8x^{9}\mathrm{d}x+\int 12x^{10}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$32\int x^{7}\mathrm{d}x+48\int x^{8}\mathrm{d}x+8\int x^{9}\mathrm{d}x+12\int x^{10}\mathrm{d}x$$

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

$$4x^{8}+48\int x^{8}\mathrm{d}x+8\int x^{9}\mathrm{d}x+12\int x^{10}\mathrm{d}x$$

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

$$4x^{8}+\frac{16x^{9}}{3}+8\int x^{9}\mathrm{d}x+12\int x^{10}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{9}\mathrm{d}x with \frac{x^{10}}{10}. Multiply 8 times \frac{x^{10}}{10}.

$$4x^{8}+\frac{16x^{9}}{3}+\frac{4x^{10}}{5}+12\int x^{10}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{10}\mathrm{d}x with \frac{x^{11}}{11}. Multiply 12 times \frac{x^{11}}{11}.

$$4x^{8}+\frac{16x^{9}}{3}+\frac{4x^{10}}{5}+\frac{12x^{11}}{11}$$

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.

$$4x^{8}+\frac{16x^{9}}{3}+\frac{4x^{10}}{5}+\frac{12x^{11}}{11}+С$$

Differentiate w.r.t. x

$$4\left(3x+2\right)\left(x^{2}+4\right)x^{7}$$