Solve ∫ (from - 1 to 2) of (-x^3+2/x)x^4=

$$\int _ { - 1 } ^ { 2 } ( - \frac { x ^ { 3 } + 2 } { x } ) x ^ { 4 } =$$

Evaluate

$$-\frac{363}{14}\approx -25.928571429$$

Solution Steps

Express \left(-\frac{x^{3}+2}{x}\right)x^{4} as a single fraction.

$$\int _{-1}^{2}\frac{-\left(x^{3}+2\right)x^{4}}{x}\mathrm{d}x$$

Cancel out x in both numerator and denominator.

$$\int _{-1}^{2}-x^{3}\left(x^{3}+2\right)\mathrm{d}x$$

Use the distributive property to multiply -x^{3} by x^{3}+2.

$$\int _{-1}^{2}-x^{6}-2x^{3}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int -x^{6}-2x^{3}\mathrm{d}x$$

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

$$-\int x^{6}\mathrm{d}x-2\int x^{3}\mathrm{d}x$$

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 -1 times \frac{x^{7}}{7}.

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

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 -2 times \frac{x^{4}}{4}.

$$-\frac{x^{7}}{7}-\frac{x^{4}}{2}$$

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.

$$-\frac{2^{7}}{7}-\frac{2^{4}}{2}-\left(-\frac{\left(-1\right)^{7}}{7}-\frac{\left(-1\right)^{4}}{2}\right)$$

Simplify.

$$-\frac{363}{14}$$