Solve ∫ (from -1 to 1) of -4x^4+4 wrt x

$$\int_{ -1 }^{ 1 } -4 { x }^{ 4 } +4 d x$$

Evaluate

$$\frac{32}{5}=6.4$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{4}\mathrm{d}x with \frac{x^{5}}{5}. Multiply -4 times \frac{x^{5}}{5}.

$$-\frac{4x^{5}}{5}+\int 4\mathrm{d}x$$

Find the integral of 4 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$-\frac{4x^{5}}{5}+4x$$

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.

$$-\frac{4}{5}\times 1^{5}+4\times 1-\left(-\frac{4}{5}\left(-1\right)^{5}+4\left(-1\right)\right)$$

Simplify.

$$\frac{32}{5}$$