Integral of an unbounded function If is unbounded

Integral of an unbounded function If is unbounded in any neighbourhood of a point then we put and continuous for a c b

and If the limits exist, we say that the integral is convergent or converges otherwise we say that it diverges or is divergent. a c b

If c = a or c = b, the integral is defined in much the same way a b

If a function F (x) exists continuous on [a, b] such that then for If so does then if converges

If that is and as then converges for m < 1 diverges for

Integrals over unbounded intervals If a function is continuous for then we put a b

If the limit exists, then we say that converges or is convergent, otherwise we say that it diverges or is divergent. By analogy, we put and

If so does then if converges

If that is and as then converges for m > 1 diverges for

Example The integral diverges.

Example

Example Determine whether the Euler-Poisson integral converges.

Put The first integral on the right-hand side is not improper and the second one is convergent since we have and

Example Determine whether the integral converges or not.

As we have Since the integral converges, so does

Example Determine whether the integral converges or not. At x = 1, the integral is discontinuous. Using the formula we obtain

This means that, as Since , we get converges, so does

LEONHARD EULER born 15 th April, 1707, Basel, Switzerland died 18 th September, 1783, St. Petersburg, Russia Swiss mathematician and physicist, one of the founders of pure mathematics. He not only made decisive and formative contributions to the subjects of geometry, calculus, mechanics, and number theory but also developed methods for solving problems in observational astronomy and demonstrated useful applications of mathematics in technology and public affairs. Euler's mathematical ability earned him the esteem of Johann Bernoulli, one of the first mathematicians in Europe at that time, and of his sons Daniel and Nicolas. In 1727 he moved to St. Petersburg, where he became an associate of the St. Petersburg Academy of Sciences and in 1733 succeeded Daniel Bernoulli to the chair of mathematics.

SIMÉON-DENIS POISSON born 21 st June, 1781, Pithiviers, France died 25 th April, 1840, Sceaux French mathematician known for his work on definite integrals, electromagnetic theory, and probability. His family coerced him into studying medicine, which he abandoned in 1798 in favour of mathematics, studying at the École Polytechnique, Paris, under the mathematicians Pierre. Simon Laplace and Joseph-Louis Lagrange, who became his lifelong friends. His life was almost entirely engaged in mathematical research and in teaching. He became a deputy professor at the École Polytechnique in 1802 and a full professor in 1806. In 1808 he was made astronomer at the Bureau des Longitudes, and, when the Faculté des Sciences was instituted in 1809, he was appointed professor of pure mathematics.