CPM Homework Help : APCALC Problem 10-46

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10-46.

Without your calculator, determine if each of the following integrals converges or diverges.

$\int _ { 1 } ^ { \infty } \frac { 1 } { x ^ { 0.9 } } d x$
Step (a):
$\lim \limits_{a\to\infty}\int_1^a\frac{1}{x^{0.9}}dx=\lim \limits_{a\to\infty}\int_{1}^{a}x^{-0.9}dx$

$\int _ { 1 } ^ { \infty } \frac { 1 } { x ^ { 1.1 } } d x$
Hint (b):
See the step in part (a).

$\int _ { 0 } ^ { 1 } \frac { 1 } { x ^ { 0.9 } } d x$
Step (c):
$\lim \limits_{c\to 0}\int_c^1\frac{1}{x^{0.9}}dx=\lim \limits_{c\to 0}\int_{c}^{1}x^{-0.9}dx$

$\int _ { 0 } ^ { 1 } \frac { 1 } { x ^ { 1.1 } } d x$
Hint (d):
See the step in part (c).

Compute without a calculator

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