Solve ∫ (from -2 to 4) of 3x^4+2 wrt x

$$\int_{ -2 }^{ 4 } 3 { x }^{ 4 } +2 d x$$

Evaluate

$$\frac{3228}{5}=645.6$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 3x^{4}+2\mathrm{d}x$$

Integrate the sum term by term.

$$\int 3x^{4}\mathrm{d}x+\int 2\mathrm{d}x$$

Factor out the constant in each of the terms.

$$3\int x^{4}\mathrm{d}x+\int 2\mathrm{d}x$$

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 3 times \frac{x^{5}}{5}.

$$\frac{3x^{5}}{5}+\int 2\mathrm{d}x$$

Find the integral of 2 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{3x^{5}}{5}+2x$$

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.

$$\frac{3}{5}\times 4^{5}+2\times 4-\left(\frac{3}{5}\left(-2\right)^{5}+2\left(-2\right)\right)$$

Simplify.

$$\frac{3228}{5}$$