Solve ∫ (from 0 to 2) of x^2k

$$\int _ { 0 } ^ { 2 } x ^ { 2 } k$$

Evaluate

$$\frac{8k}{3}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{2}k\mathrm{d}x$$

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

$$k\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}.

$$k\times \frac{x^{3}}{3}$$

Simplify.

$$\frac{kx^{3}}{3}$$

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.

$$\frac{1}{3}k\times 2^{3}-\frac{1}{3}k\times 0^{3}$$

Simplify.

$$\frac{8k}{3}$$

Differentiate w.r.t. k

$$\frac{8}{3} = 2\frac{2}{3} = 2.6666666666666665$$