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Problem 14 Easy Difficulty

Evaluate the integral.

$ \displaystyle \int \frac{\sin^2 \left(\frac{1}{t} \right)}{t^2} dt $


$\frac{1}{4} \sin (2 / t)-\frac{1}{2 t}+C$


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Video Transcript

this problem is from Chapter seven section to problem number fourteen in the book Calculus Early Transcendence. ALS eighth edition, My James Door Here we have a indefinite integral of sine squared of one over t all divided by t squared. So before we start integrate here, we can apply u substitution. Let's take you two be one over tea so that to you using the powerful from differential calculus is negative. One over T square dt equivalently negative to you is one over teeth clear. Titi. So if we apply our u substitution here, we can rewrite our original integral as negative in a girl Signed square of you Deal Now we can apply an identity for science where using a trick in a metric identity. Let's re write this expression here the inside rent We have integral negative in general and we can rewrite this sign square as one minus co sign off to you all divided by two. And now we could integrate each of these two terms And for the second term am I helped to use another use of and the integration you could take. Let's stay here W since we're already using you for a previous use up here. You could use another letter w for your other u sub and you could take that to be to you so integrating each of these terms. So in a girl, one over two becomes you over, too. And for the second, integral, we have a negative sign of to you over two times two, which is for and our constant of integration seat. So we have two steps left. We can distribute the negative sign and through the parentheses. And we could also replace you with one oversee. So, doing both of these Ted step simultaneously. We have negative one over Tootie, plus sign of two over tea over floor, plus see, and that's your final answer.