Question

7–58. Improper integrals Evaluate the following integrals or state that they diverge.47. ∫ (from 0 to 10) 1/∜(10 - x) dx

Accepted Answer

Identify the integral and the function to be integrated: $$ \int_0^{10} \frac{1}{\sqrt[4]{10 - x}} \, dx . $$Notice that the integrand involves a fourth root in the denominator, which can cause issues near $$ x = 10 . $$Check for any points of discontinuity or where the integrand might be undefined within the interval $$[0, 10]. $$Since $$ \sqrt[4]{10 - x} $$ becomes zero at $$ x = 10 $$, the integrand tends to infinity there, indicating an improper integral at the upper limit. Rewrite the integral as a limit to handle the improper behavior at $$ x = 10 $$: $$ \lim_{t 	o 10^-} \int_0^t \frac{1}{(10 - x)^{1/4}} \, dx . $$This allows us to evaluate the integral up to a point $$ t \u003c 10 $$ and then take the limit as $$ t $$ approaches 10 from the left. Perform a substitution to simplify the integral. Let $$ u = 10 - x $$, which implies $$ du = -dx . $$Change the limits accordingly: when $$ x = 0 $$, $$ u = 10 $$; when $$ x = t $$, $$ u = 10 - t . $$The integral becomes $$ \int_{u=10}^{u=10 - t} u^{-1/4} (-du) = \int_{10 - t}^{10} u^{-1/4} \, du . $$Integrate $$ u^{-1/4} $$ with respect to $$ u $$ using the power rule for integrals: $$ \int u^n \, du = \frac{u^{n+1}}{n+1} + C $$, where $$ n = -\frac{1}{4} . $$After integrating, substitute back the limits $$ u = 10 - t $$ and $$ u = 10 $$, then take the limit as $$ t 	o 10^-  to $$determine if the integral converges or diverges.

7–58. Improper integrals Evaluate the following integrals or state that they diverge.
47. ∫ (from 0 to 10) 1/∜(10 - x) dx

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Key Concepts

Improper Integrals

Behavior of the Integrand Near Critical Points

Evaluating Limits in Definite Integrals