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

$$\int _ { - 4 } ^ { 2 } ( 4 x ^ { 4 } + 3 x ^ { 3 } ) d x$$

Evaluate

$$\frac{3324}{5}=664.8$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

$$\frac{4x^{5}}{5}+3\int x^{3}\mathrm{d}x$$

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 3 times \frac{x^{4}}{4}.

$$\frac{4x^{5}}{5}+\frac{3x^{4}}{4}$$

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.

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

Simplify.

$$\frac{3324}{5}$$