Find out the answer to "Determine whether the improper integral diverges or converges...."

Brittney Lord

Answered question

2021-11-05

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges.

\int_{0}^{5} \frac{1}{25 - x^{2}} d x

comentezq

Skilled2021-11-06Added 106 answers

Step 1
The given integral is

\int_{0}^{5} \frac{1}{25 - x^{2}} d x

.
Check the convergence as follows.
Step 2

\int_{0}^{5} \frac{1}{25 - x^{2}} d x = \int_{0}^{5} \frac{1}{5^{2} - x^{2}} d x

= {[\frac{1}{2 (5)} \ln (\frac{5 + x}{5 - x})]}_{0}^{5}

= \frac{1}{10} [\ln (\frac{5 + 5}{5 - 5}) - \ln (\frac{5 + 0}{5 - 0})]

= \frac{1}{10} (\ln (\infty) - 0)

= \infty

Thus, the integral

\int_{0}^{5} \frac{1}{25 - x^{2}} d x

diverges.

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