Solve ∫ (from 2 to 8) of 1/4x^3 wrt x

$$\int _ { 2 } ^ { 8 } \frac { 1 } { 4 } x ^ { 3 } d x$$

Evaluate

$$255$$

Solution Steps

Evaluate the indefinite integral first.

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

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

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

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}.

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

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.

$$\frac{8^{4}}{16}-\frac{2^{4}}{16}$$

Simplify.

$$255$$