Solve ∫ (from 1 to 100) of 4/n^3

$$\int _ { 1 } ^ { 100 } \frac { 4 } { n ^ { 3 } }$$

Evaluate

$$\frac{396}{n^{3}}$$

Solution Steps

Evaluate the indefinite integral first.

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

Find the integral of \frac{4}{n^{3}} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{4}{n^{3}}x$$

Simplify.

$$\frac{4x}{n^{3}}$$

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.

$$4n^{-3}\times 100-4n^{-3}$$

Simplify.

$$\frac{396}{n^{3}}$$

Differentiate w.r.t. n

$$-\frac{1188}{n^{4}}$$