Solve ∫ (from 0 to 2) of x(n^2-3)8/27

$$\int _ { 0 } ^ { 2 } x ( n ^ { 2 } - 3 ) \frac { 8 } { 27 }$$

Evaluate

$$\frac{16\left(n^{2}-3\right)}{27}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{8x\left(n^{2}-3\right)}{27}\mathrm{d}x$$

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

$$\frac{8\left(n^{2}-3\right)}{27}\int x\mathrm{d}x$$

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

$$\frac{8\left(n^{2}-3\right)}{27}\times \frac{x^{2}}{2}$$

Simplify.

$$\frac{4\left(n^{2}-3\right)x^{2}}{27}$$

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.

$$\frac{4}{27}\left(n^{2}-3\right)\times 2^{2}-\frac{4}{27}\left(n^{2}-3\right)\times 0^{2}$$

Simplify.

$$\frac{16n^{2}}{27}-\frac{16}{9}$$

Differentiate w.r.t. n

$$\frac{32n}{27}$$