Solve ∫ (from 1 to 3) of 100/x^3 wrt x

$$\int_{ 1 }^{ 3 } \frac{ 100 }{ { x }^{ 3 } } d x$$

Evaluate

$$\frac{400}{9}\approx 44.444444444$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{100}{x^{3}}\mathrm{d}x$$

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

$$100\int \frac{1}{x^{3}}\mathrm{d}x$$

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 100 times -\frac{1}{2x^{2}}.

$$-\frac{50}{x^{2}}$$

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.

$$-50\times 3^{-2}+50\times 1^{-2}$$

Simplify.

$$\frac{400}{9}$$