Home page for accesible maths 1 1 Further Integration 1.50 Example of an integral over the real line 1.52 Integrals of unbounded functions

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1.51 Another integral over the real line

Example.

The integral $\int_{-\infty}^{\infty}x\,dx$ diverges.

Solution. We have

\int_{0}^{R}x\,dx={{{1}\over{2}}R^{2}\rightarrow\infty}

as $R\rightarrow\infty$ ; so $\int_{0}^{\infty}x\,dx$ diverges. Clearly $\int_{-R}^{R}x\,dx={0,}$ but the integral nevertheless diverges because there are infinite areas above and below the $x$ -axis.