Evaluate: integral of _{} 0 ^ b x ^ 3 x

Expression: $$\int _ { 0 } ^ { b } x ^ { 3 } d x$$

Evaluate the indefinite integral first.

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

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

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

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.

$$\frac{b^{4}}{4}-\frac{0^{4}}{4}$$

Simplify.

$$\frac{b^{4}}{4}$$