Solve ∫ (from - 11 to 1) of (-13x^3+3x^7+87) wrt x

$$\int _ { - 11 } ^ { 1 } ( - 13 x ^ { 3 } + 3 x ^ { 7 } + 87 ) d x$$

Evaluate

$$-80335956$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -13x^{3}+3x^{7}+87\mathrm{d}x$$

Integrate the sum term by term.

$$\int -13x^{3}\mathrm{d}x+\int 3x^{7}\mathrm{d}x+\int 87\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-13\int x^{3}\mathrm{d}x+3\int x^{7}\mathrm{d}x+\int 87\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. Multiply -13 times \frac{x^{4}}{4}.

$$-\frac{13x^{4}}{4}+3\int x^{7}\mathrm{d}x+\int 87\mathrm{d}x$$

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

$$-\frac{13x^{4}}{4}+\frac{3x^{8}}{8}+\int 87\mathrm{d}x$$

Find the integral of 87 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$-\frac{13x^{4}}{4}+\frac{3x^{8}}{8}+87x$$

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.

$$-\frac{13}{4}\times 1^{4}+\frac{3}{8}\times 1^{8}+87\times 1-\left(-\frac{13}{4}\left(-11\right)^{4}+\frac{3}{8}\left(-11\right)^{8}+87\left(-11\right)\right)$$

Simplify.

$$-80335956$$