MAC 2312 Lecture 9: 7.8 improper integrals

MATH 31S In order to compute the above quantities, one needs to be able to evaluate infinite inte- grals. We use the following approach. For any real number a define f(x)dx= linn | f(z)dz. when the limit exists and is finite, then the integral is said to be convergent, otherwise it is divergent. Similarly f(x)dェ=tlinnol, f(x)dx. (2) Problem 3 Determine whether the following integrals are convergent or divergent. If -00 convergent, determine their value: dr,dr cos(x)dr. (3) Problem 4 Dctcrminc for which p >0, the integral is convergent and divergent.

For each of the improper integrals below, if the comparison test applies, enter either A or B followed by one letter from C to K that best applies, and if the comparison test does not apply, enter only L. For example, one possible answer is BF, and another one is L. 1. integral^infinity_1 cos^2 (x)/x^2 + 3 dx 2. integral^infinity_1 10 + sin (x)/squareroot x - 0.2 dx 3. integral^infinity_1e^-x/x^2 dx 4. integral^infinity_1x/squareroot x^6 + 3 dx 5. integral^infinity_1 1/x^2 + 3 dx A. The integral is convergent B. The integral is divergent C. by comparison to integral^infinity_1 1/x^2 - 3 dx. D. by comparison to integral^infinity_1 1/x^2 + 3 dx. E. by comparison to integral^infinity_1 cos^2 (x)/x^2 dx. F. by comparison to integral^infinity_1 e^x/x^2 dx. G. by comparison to integral^infinity_1 -e^-x/2x dx. H. by comparison to integral^infinity_1 1/squareroot x dx. I. by comparison to integral^infinity_1 1/squareroot x^5 dx. J. by comparison to integral^infinity_1 1/x^2 dx. K. by comparison to integral^infinity_1 1/x^3 dx. L. The comparison test does not apply.