Step 1

The objective is to evaluate the given improper integrals:

\(\int_{1}^{\infty}\frac{1}{x^{3}}dx\)

Step 2

Evaluate the given integral as:

\(\int_{1}^{\infty}\frac{1}{x^{3}}dx=[\int\frac{1}{x^{3}}dx]_{1}^{\infty}\)

\(=[\frac{x^{-3+1}}{-3+1}]_{1}^{\infty}=[\frac{1}{-2x^{2}}]_{1}^{\infty}=[0+\frac{1}{2\times 1^{2}}]\)

\(=\frac{1}{2}\)

Thus,

\(\int_{1}^{\infty}\frac{1}{x^{3}}dx=\frac{1}{2}\)

The objective is to evaluate the given improper integrals:

\(\int_{1}^{\infty}\frac{1}{x^{3}}dx\)

Step 2

Evaluate the given integral as:

\(\int_{1}^{\infty}\frac{1}{x^{3}}dx=[\int\frac{1}{x^{3}}dx]_{1}^{\infty}\)

\(=[\frac{x^{-3+1}}{-3+1}]_{1}^{\infty}=[\frac{1}{-2x^{2}}]_{1}^{\infty}=[0+\frac{1}{2\times 1^{2}}]\)

\(=\frac{1}{2}\)

Thus,

\(\int_{1}^{\infty}\frac{1}{x^{3}}dx=\frac{1}{2}\)