Solve ∫ 90x^8(1-x)

$$\int 90 x ^ { 8 } ( 1 - x )$$

Evaluate

$$С+10x^{9}-9x^{10}$$

Solution Steps

Use the distributive property to multiply 90x^{8} by 1-x.

$$\int 90x^{8}-90x^{9}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 90x^{8}\mathrm{d}x+\int -90x^{9}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$90\int x^{8}\mathrm{d}x-90\int x^{9}\mathrm{d}x$$

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

$$10x^{9}-90\int x^{9}\mathrm{d}x$$

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 -90 times \frac{x^{10}}{10}.

$$10x^{9}-9x^{10}$$

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.

$$10x^{9}-9x^{10}+С$$

Differentiate w.r.t. x

$$90\left(1-x\right)x^{8}$$