Solve ∫ x^7(2x+7) wrt x

$$\int x ^ { 7 } ( 2 x + 7 ) d x$$

Evaluate

$$\frac{2x^{9}}{9}+\frac{7x^{8}}{8}+С$$

Solution Steps

Use the distributive property to multiply x^{7} by 2x+7.

$$\int 2x^{8}+7x^{7}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2x^{8}\mathrm{d}x+\int 7x^{7}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x^{8}\mathrm{d}x+7\int x^{7}\mathrm{d}x$$

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

$$\frac{2x^{9}}{9}+7\int x^{7}\mathrm{d}x$$

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

$$\frac{2x^{9}}{9}+\frac{7x^{8}}{8}$$

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.

$$\frac{2x^{9}}{9}+\frac{7x^{8}}{8}+С$$

Differentiate w.r.t. x

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