Solve ∫ x^6(5x-6) wrt x

$$\int x ^ { 6 } ( 5 x - 6 ) d x$$

Evaluate

$$\frac{5x^{8}}{8}-\frac{6x^{7}}{7}+С$$

Solution Steps

Use the distributive property to multiply x^{6} by 5x-6.

$$\int 5x^{7}-6x^{6}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 5x^{7}\mathrm{d}x+\int -6x^{6}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$5\int x^{7}\mathrm{d}x-6\int x^{6}\mathrm{d}x$$

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

$$\frac{5x^{8}}{8}-6\int x^{6}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{6}\mathrm{d}x with \frac{x^{7}}{7}. Multiply -6 times \frac{x^{7}}{7}.

$$\frac{5x^{8}}{8}-\frac{6x^{7}}{7}$$

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.

$$\frac{5x^{8}}{8}-\frac{6x^{7}}{7}+С$$

Differentiate w.r.t. x

$$\left(5x-6\right)x^{6}$$