Solve ∫ (from 1 to 100) of 4/x^3

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

Evaluate

$$\frac{9999}{5000}=1.9998$$

Solution Steps

Evaluate the indefinite integral first.

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

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

$$4\int \frac{1}{x^{3}}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int \frac{1}{x^{3}}\mathrm{d}x with -\frac{1}{2x^{2}}. Multiply 4 times -\frac{1}{2x^{2}}.

$$-\frac{2}{x^{2}}$$

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.

$$-2\times 100^{-2}+2\times 1^{-2}$$

Simplify.

$$\frac{9999}{5000}$$