Solve ∫ (from - 100 to 251) of (x^5+6x-7x^8) wrt x

$$\int _ { - 100 } ^ { 251 } ( x ^ { 5 } + 6 x - 7 x ^ { 8 } ) d x$$

Evaluate

$$-\frac{6152602105682061900273}{2}=-3.076301053 \cdot 10^{21}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{5}+6x-7x^{8}\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{5}\mathrm{d}x+\int 6x\mathrm{d}x+\int -7x^{8}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{5}\mathrm{d}x+6\int x\mathrm{d}x-7\int x^{8}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{5}\mathrm{d}x with \frac{x^{6}}{6}.

$$\frac{x^{6}}{6}+6\int x\mathrm{d}x-7\int x^{8}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply 6 times \frac{x^{2}}{2}.

$$\frac{x^{6}}{6}+3x^{2}-7\int x^{8}\mathrm{d}x$$

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 -7 times \frac{x^{9}}{9}.

$$\frac{x^{6}}{6}+3x^{2}-\frac{7x^{9}}{9}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{251^{6}}{6}+3\times 251^{2}-\frac{7}{9}\times 251^{9}-\left(\frac{\left(-100\right)^{6}}{6}+3\left(-100\right)^{2}-\frac{7}{9}\left(-100\right)^{9}\right)$$

Simplify.

$$-\frac{6152602105682061900273}{2}$$