Integrals and limits

### 12.3.7 Integrals and limits

The limit command (see Section 12.1.1) can compute limits involving integrals.

#### Examples

Find the limit of ∫2a x−2 dx as a→+∞.

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

limit(integrate(1/(x^2),x,2,a),a,+(infinity))

1
2

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

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

limit(integrate(x/(x^2-1)+log((x+1)/(x-1)),x,2,a),a,+infinity)

+∞ |  |  |  |  |  |  |  |  |  |

Indeed, since ∫2a x/(x2−1) dx=(1/2)(ln(a2−1)−ln(3)) and ∫2aln((x+1)/(x−1)) dx=ln(a+1)+ln(a−1) +aln((a+1)/(a−1))−3ln(3), the integral ∫2ax/(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 ∫a3acos(x)/x dx as a→ 0.

limit(int(cos(x)/x,x,a,3a),a,0)

ln | ⎛ ⎝ | 3 | ⎞ ⎠

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 ∫a3a(1/x−x/2) dx ≤ ∫a3acos(x)/x dx ≤ ∫a3a1/x dx, which gives you ln(3)−2a2 ≤ ∫a3acos(x)/x dx ≤ ln(3), and so as a approaches 0, ∫a3acos(x)/x dx will approach ln(3).