Solve ∫ (2x^7+4x-9) wrt x

$$\int{ (2 { x }^{ 7 } +4x-9) }d x$$

Evaluate

$$\frac{x^{8}}{4}+2x^{2}-9x+С$$

Solution Steps

Integrate the sum term by term.

$$\int 2x^{7}\mathrm{d}x+\int 4x\mathrm{d}x+\int -9\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x^{7}\mathrm{d}x+4\int x\mathrm{d}x+\int -9\mathrm{d}x$$

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

$$\frac{x^{8}}{4}+4\int x\mathrm{d}x+\int -9\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply 4 times \frac{x^{2}}{2}.

$$\frac{x^{8}}{4}+2x^{2}+\int -9\mathrm{d}x$$

Find the integral of -9 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{8}}{4}+2x^{2}-9x$$

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.

$$\frac{x^{8}}{4}+2x^{2}-9x+С$$

Differentiate w.r.t. x

$$2x^{7}+4x-9$$