Solve ∫ (from 1 to 20) of x^3/2 wrt x

$$\int _ { 1 } ^ { 20 } \frac { x ^ { 3 } } { 2 } d x$$

Evaluate

$$\frac{159999}{8}=19999.875$$

Solution Steps

Evaluate the indefinite integral first.

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

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

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

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}.

$$\frac{x^{4}}{8}$$

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.

$$\frac{20^{4}}{8}-\frac{1^{4}}{8}$$

Simplify.

$$\frac{159999}{8}$$