Question

\(5 - 32\)Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

9. \(\int_2^\infty {{e^{ - 5p}}} dp\)

Short answer

Integral is convergent & converges to\(\frac{{{e^{ - 10}}}}{5}\).

Step-by-step solution

Definition for converging integral

Consider the integral\(\int_{\bf{a}}^{\bf{t}} {\bf{f}} {\bf{(x)dx}}\)is for the limit with the finite number\(t \ge a\). Then, the integral is written as:

\(\int_a^\infty f (x)dx = \mathop {\lim }\limits_{t \to \infty } \int_a^t f (x)dx\)

Evaluate the integral

In order to determine if integral converges or diverges we evaluate both integrals i.e; \(\int_a^\infty f (x)dx\,\& \,\mathop {\lim }\limits_{t \to \infty } \int_a^t f (x)dx\)& see if both integrals are equal or not.

\(\begin{aligned}{c}\int_2^\infty {{e^{ - 5p}}} dp &= \mathop {\lim }\limits_{t \to \infty } \int_2^t {{e^{ - 5p}}} dp\;\\ &= \mathop {\lim }\limits_{t \to \infty } \left( {\frac{{{e^{ - 5p}}}}{{ - 5}}} \right)_2^t\;\;\;{\rm{ Use }}\int {{e^{ax}}} dx &= \frac{{{e^{ax}}}}{a} + C.\;\;\;\\ &= \mathop {\lim }\limits_{t \to \infty } \left( {\frac{{{e^{ - 5t}}}}{{ - 5}} - \frac{{{e^{ - 5 \cdot 2}}}}{{ - 5}}} \right)\\ &= \mathop {\lim }\limits_{t \to \infty } \left( { - \frac{{{e^{ - 5t}}}}{5}} \right) + \frac{{{e^{ - 10}}}}{5}\end{aligned}\)

Solve further as.

\(\begin{aligned}{c}\int_2^\infty {{e^{ - 5p}}} dp &= - \frac{0}{5} + \frac{{{e^{ - 10}}}}{5}\\ &= \frac{{{e^{ - 10}}}}{5}\end{aligned}\)

The limit exists as a finite number and so the given integral is convergent.