Home page for accesible maths 1 1 Further Integration

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1.58 Comparison Test.

Example.

The following integral converges:

I=∫1∞sin⁡xx2⁢d⁢x.I=\int_{1}^{\infty}{{\sin x}\over{x^{2}}}dx.

Solution. Here |sin⁡x|≤1|\sin x|\leq 1, and hence |sin⁡x|x2≤1x2.{{|\sin x|}\over{x^{2}}}\leq{{{1}\over{x^{2}}}.} Also, ∫1∞d⁢xx2\int_{1}^{\infty}\frac{dx}{x^{2}} converges since

∫1Rd⁢xx2=[-1x]1R=1-1R→1  as R→∞R\rightarrow\infty.\int_{1}^{R}{{dx}\over{x^{2}}}={\left[{{-1}\over{x}}\right]_{1}^{R}}\,{=1-{{1}% \over{R}}}\,{\rightarrow 1}\qquad\mbox{as $R\rightarrow\infty$}.

Hence II converges and has absolute value at most 11 by the comparison test. However, we don’t have a simple formula for the value of the integral.