In Exercises 39–48, use an appropriate substitution and then a...

Question

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.

∫ (e^{t} dt) / ((1 + e^{2t})^{3/2}) from ln(3/4) to ln(4/3)

Accepted Answer

Welcome back everyone. In this problem we want to evaluate the integral of 6 e x divided by 1 plus e 2x the power of 3 halves with respect to x between the limits of the ln of 5/12 and the ln of 12/5. A says it's 7/13, B42/13, C 30-13, and D 72/13. Now for us to evaluate this integral, let's first try to rewrite it so that we can get our limits in a simpler form. So by substitution in this case, OK. Let's let you be equal to eat the eggs. In that case, the derivative of u with respect to x would be equal to e x, and that means duu then would be equal to e x dx. On the other hand, If u equals e to the x, then x would be equal to the natural log or ln of 5/12. And if that's the case, that means you would be equal to e ln of 5/12, which would be equal to 5/12. Likewise, if x is equal to the ln of 12/5, if we apply the same logic, u would then be equal to e to the ln of 12/5, which would be equal to 12/5 by the properties of logs. So by substituting this into our integral, this means that our integral or original integral that I can rewrite here, OK. So our original integral. Would be equal to 6 multiplied by the integral of between the limits of 5/12 and 12/5 of d u divided by 1 plus u squared to the power of 3 halves. Now it makes it a little bit easier to integrate, but again we can use some substitution to simplify it even further. Now by substitution, again, trigonometric substitution in this case, let's set you. To be equal to the tangent of theta. This means the derivative of u with respect to theta would be sex squared theta, so that means d u would be equal to sex squared theta d theta, and that will help us to simplify the denominator of our integrant, so. Before, but, however, before we write that, we can also rewrite our limits, where since we know that you equals the tangent of theta, that means that theta would be equal to the inverse tangent of you. OK, so if you equals 5/12. OK, then theta would be equal to the inverse tangent of 5/12. And if u equals 12/5, then theta equals the inverse tangent of 12/5. So now that we have the new limits and or substitutions, then this means that we could rewrite 6 multiplied by the integral between 5/12 and 12/5 of d u divided by 1 plus u 2 to the power of 3 halves. As the integral between the limits of the inverse tangent of 5/12, OK, and the inverse tangent of 12/5s, OK. Of Sick, but let me write it out here. square theta d theta divided by se cubed theta and in this case we could simplify it even more to now get our integral, OK. To know get our integral. Of The theta divided by sec theater or cost theater, which is the, the reciprocal of sec theta d theta, so it's really a definite integral of cost theater with respect to theta. No, where, where do we go from here? Well, if we apply our limits, yeah. Then this is going to be 6 multiplied by the sine of theta between the bones of the inverse tangent of 5/12 and the inverse tangent of 12/5, OK. Now how can we evaluate these because we know that this is what we would substitute into theta. Well, notice here that since the, sorry, yeah, since tan theta equals u, if we visualize this as a triangle, OK, then that means the opposite side. So, but rather let's draw our right triangle here. If we visualize this as a right triangle with our angle theta, then our opposite side would be you, OK, and the adjacent side would be 1. So that means the hypotenuse would be the square root of the sums of the squares of both sides. So that's the square root of 1 plus u squared. So since sine theta equals u, then this means sine theta. Which would be equal to the sine of the inverse tangent of u would be equal to u divided by the square root of 1 + u square. So really then if we were to apply. Our bone juice here, yeah, our our bones. Then that means we could rewrite this as, well, let me write it out for first here, so 6 sine theta between our bones. OK. Would be equal to 6 multiplied by the sign. Of the inverse tangent of 12/5, OK. Minus sine multiplied by the inverse tangent. Of 5/12 And using what we know, what we've, what we've just discovered here, that means this would be equal to. 12/5 divided by the square root of 1 + 12/5 squared, OK, minus 5/12. OK, of the square root of 1 + 5/12 squared. I know if we simplify here, and we're also multiplying all of this by 6, my apologies, let me just write that in here. Yeah I know when we substitute here when when we evaluate this, OK, it ends up being 12/5 divided by 13/5 or 12/13. And for a second term it ends up being 5/12 divided by 13/12, which is 5/13. 12/13 minus 5/13 is 7/13, and when we multiply that by 6, we end up with 42/13. Therefore, our definite integral is 42/13. Therefore, B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.∫ (e^{t} dt) / ((1 + e^{2t})^{3/2}) from ln(3/4) to ln(4/3)

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In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ (e^{t} dt) / ((1 + e^{2t})^{3/2}) from ln(3/4) to ln(4/3)