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Show how use IBP t0 integrale: sec (0)40Use Trig Substitution t0 Integrate: [rv4+9x" dx...

Question

Show how use IBP t0 integrale: sec (0)40Use Trig Substitution t0 Integrate: [rv4+9x" dx

Show how use IBP t0 integrale: sec (0)40 Use Trig Substitution t0 Integrate: [rv4+9x" dx



Answers

First make a substitution and then use integration by parts to evaluate the integral.

$ \displaystyle \int_0^\pi e^{\cos t} \sin 2t dt $

Evasive. Using the Trigana metric identity, the sign of two theater is equal to two times the silent state attempt Coast elevator. We can be redder integral as the integral from zero to pie off to co sign of t e to the power off the coastline of tea Times this sign off T d t. So we're gonna let c is equal to the coast side of tea So that negative easy is it to the sine of t the sign of t d t. So once we apply the substitution, we basically end up with the integral off negative to Z E to the power of Z d. C. So again, we're gonna use use up your the substitutions they use equal to negative to Z so that d'you is equal to negative. Too easy. Devi is then equal to G to the power of Z t Z and the is equal to eat up our Z using the integration by parts formula, we can end up with negative Tuesay negative to Z E to the power of Z plus. We can pull the two out of the integral of each of the power of Z d Z So once we integrate, we end up with that. This is equal to negative, too. On then, of course, back substituting. We end up with negative to co sign of tea coasts. I t e to the power off the coast of P plus two e to the power off the coast side, off t plus c So we have to evaluate this from zero to pie. So once we plug in numbers and apply the fundamental fear of capitalists, we end up with two e to the power of negative one plus two to the power of negative one minus negative. To me, to the power of one plus two beat the power of one. So this is just equal to four e to the power of one.

The problem with Give the power fine thanks into sign 12 digs. Mr Paul, You Science Politics equals Coptics. The people are these sound wins the X people Did they? Therefore we have a lower limit. Equal Do can be this one. You get the power. Wow. So this is our answer.

What's so to evaluate girl? You're a metric space to grow coach time Expedia. Yeah. Rewrite this as one very negative ones. X. Co size good Plus 1 30 times to integrate integrate of course I two X. Yes. The softest black. Let's see the equal to using substitution that you meet tonight. Indeed. Feel the course through Coulson. So you have Maybe have one of the three. Yeah. Close sign to react Plus 1/3 time. I minus one. He said the T. V. So keep the queue. It's up to him. Back teeth was signed. Tv comes size three at close your eyes. Mhm.

Okay, This question wants us to integrate the following function. So since we're dealing with trig functions here, we should probably use our U substitution in our argument with that trick function. So let's pick you equal to two tea. So this means that d u is just two times d t. But we don't have to d t anywhere in our integral we only have one d t. So let's divide both sides of that equation by two to get do you over two equals DT So now we can plug in and get that are integral is now just the integral c Can't you tangent you times do you over too or 1/2 times the integral of C Can't you tangent you? Do you and this anti derivative right here is just one of our basic ones for Trig It's the anti derivative which just gives us He can't because the derivative of seeking seeking tangent. So this is just 1/2 see, can't you plus c or 1/2 c can't of to t plus c when we substitute back in for you


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