Solve ∫ x^8(5x-3) wrt x

$$\int x ^ { 8 } ( 5 x - 3 ) d x$$

Evaluate

$$\frac{x^{10}}{2}-\frac{x^{9}}{3}+С$$

Solution Steps

Use the distributive property to multiply x^{8} by 5x-3.

$$\int 5x^{9}-3x^{8}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 5x^{9}\mathrm{d}x+\int -3x^{8}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$5\int x^{9}\mathrm{d}x-3\int x^{8}\mathrm{d}x$$

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 5 times \frac{x^{10}}{10}.

$$\frac{x^{10}}{2}-3\int x^{8}\mathrm{d}x$$

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 -3 times \frac{x^{9}}{9}.

$$\frac{x^{10}}{2}-\frac{x^{9}}{3}$$

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.

$$\frac{x^{10}}{2}-\frac{x^{9}}{3}+С$$

Differentiate w.r.t. x

$$\left(5x-3\right)x^{8}$$