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(a) If $g(x) = \frac{(\sin^2 x)}{x^2}$, use your calculator or computer to make a table of approximate values of $\displaystyle \int_1^t g(x)\ dx$ for $t = 2, 5, 10, 100, 1000$, and $10,000$. Does it appear that $\displaystyle \int_1^\infty g(x)\ dx$ is convergent?(b) Use the Comparison Theorem with $f(x) = \frac{1}{x^2}$ to show that $\displaystyle \int_1^\infty g(x)\ dx$ is convergent.(c) Illustrate part (b) by graphing $f$ and $g$ on the same screen for $1 \le x \le 10$. Use your graph to explain intuitively why $\displaystyle \int_1^\infty g(x)\ dx$ is convergent.

a. $g(x)=\frac{\sin ^{2} x}{x^{2}}$b. $\int_{1}^{\infty} \frac{\sin ^{2} x}{x^{2}} d x$ is convergentc. convergent

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Catherine R.

Missouri State University

Anna Marie V.

Campbell University

Boston College

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Video Transcript

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Integration Techniques

Catherine R.

Missouri State University

Anna Marie V.

Campbell University

Boston College

Lectures

Join Bootcamp