13.3.7  Integrals and limits

The limit command (see Section 13.1.1) can compute limits involving integrals.

Examples

Find the limit of ∫₂^a x⁻² dx as a→+∞.

Input (if a is assigned, first input purge(a)):

Indeed, since ∫₂^ax⁻² dx=1/2−1/a, the integral ∫₂^ax⁻² dx tends to 1/2 as a goes to infinity.

Find the limit of ∫₂^a (x/x²−1+ln(x+1/x−1))dx as a→+∞. (If a is assigned, first input purge(a).)

Indeed, since ∫₂^a x/(x²−1) dx=(1/2)(ln(a²−1)−ln(3)) and ∫₂^aln((x+1)/(x−1)) dx=ln(a+1)+ln(a−1) +aln((a+1)/(a−1))−3ln(3), the integral ∫₂^ax/(x²−1)+ln((x+1)/(x−1)) dx goes to infinity as a goes to infinity.

For an example when the integral cannot be simply evaluated, find the limit of ∫_a^3acos(x)/x dx as a→ 0.

To find this limit yourself, you cannote that 1−x²/2 ≤ cos(x) ≤ 1, and so 1/x−x/2 ≤ cos(x)/x ≤ 1/x, and so ∫_a^3a(1/x−x/2) dx ≤ ∫_a^3acos(x)/x dx ≤ ∫_a^3a1/x dx, which gives you ln(3)−2a² ≤ ∫_a^3acos(x)/x dx ≤ ln(3), and so as a approaches 0, ∫_a^3acos(x)/x dx will approach ln(3).