Solve ∫ (from - 1 to 3) of (5-1/4x^3) wrt x

$$\int _ { - 1 } ^ { 3 } ( 5 - \frac { 1 } { 4 } x ^ { 3 } ) d x$$

Evaluate

$$15$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

Find the integral of 5 using the table of common integrals rule \int a\mathrm{d}x=ax.

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

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 -\frac{1}{4} times \frac{x^{4}}{4}.

$$5x-\frac{x^{4}}{16}$$

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.

$$5\times 3-\frac{3^{4}}{16}-\left(5\left(-1\right)-\frac{\left(-1\right)^{4}}{16}\right)$$

Simplify.

$$15$$