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1.53 Integrals of unbounded functions

For example, the integral ∫01log⁡x⁢d⁢x\int_{0}^{1}\log x\,dx is improper since log⁡x\log x diverges at x=0x=0. But

∫δ1log⁡x⁢d⁢x=[x⁢log⁡x-x]δ1=-1-δ⁢log⁡δ+δ.\int_{\delta}^{1}\log x\,dx={\left[x\log x-x\right]_{\delta}^{1}=}\,{-1-\delta% \log\delta+\delta.}

Since δ⁢log⁡δ→0\delta\log\delta\rightarrow 0 as δ→0+\delta\rightarrow 0+ (see frame 4.23 in MATH101), the integral converges to -1-1 as δ→0+\delta\rightarrow 0+.

Likewise, if ff is discontinuous at bb, then we can integrate over (a,b-δ)(a,b-\delta) and define

∫abf⁢(x)⁢d⁢x=limδ→0+⁡∫ab-δf⁢(x)⁢d⁢x\int_{a}^{b}f(x)\,dx=\lim_{\delta\rightarrow 0+}\int_{a}^{b-\delta}f(x)\,dx

where this limit exists; otherwise, we say that ∫abf⁢(x)⁢d⁢x\int_{a}^{b}f(x)\,dx diverges.