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Consider the integral Jr"e" dz: Applying the integration by parts technique , letanddvdxThen UUIvdu =dx...

Question

Consider the integral Jr"e" dz: Applying the integration by parts technique , letanddvdxThen UUIvdu =dx

Consider the integral Jr"e" dz: Applying the integration by parts technique , let and dv dx Then UU Ivdu = dx



Answers

Use integration by parts to evaluate the given integral. $$ \int e^{\alpha x} \sin \beta x d x $$

To integrate this, we will apply integration by parts and here you want to let you be the expression that cannot be integrated and so that will be tangent inverse of x. And devi will be the remaining factors inside this instagram that will be x squared dx. Now to find the differential a view, we simply take the differential of both sides of this equation and that will be do you equal to 1/1 plus x squared dx. And to get V we simply take the integral of both sides of this and we get the equal to x rays to the third power over three. And so by integration by parts, this is equal to U times v minus the integral of E D U. That will be tangent inverse of X times V which is x cube over three minus the integral of x cube over three times D U which is one over one plus x squared dx. Now simplifying this, we have x cubed times tangent inverse of x over three minus 1/3 integral of X cube over one plus x squared dx. Now for this integral we will apply substitution and here we want to let W equal to one plus x squared. And so the W is just equal to two X dx. Or that means one half dws X dx. Now, since the numerator is x rays to the third power and we already have X dx in terms of W. Then we need to find an expression for X squared in terms of W by manipulating this questions. So we get x squared equal to W -1. And so from here we have x rays through the third power times tangent inverse of X over three -1/3 times the integral of We have w minus one Times 1/2 dw this all over W which is equal to x cubed times tangent inverse of x over three -1/6 times 1 -1 over WDW. And so integrating this term by term, we have x cubed times tangent inverse of x Over 3 -1/6 times W minus L and absolute value of W. And then plus C. And because W is one plus X squared, then we have X cubed times tangent inverse of X over three minus 1/6 times one plus x squared minus Ln absolute value of one plus x squared or without the absolute value since one plus x ray is always positive and then plus C and Distributing the 1/6. We have X cubed times tangent inverse of X over three -1/6 minus X squared over six plus 1/6 Ln of one plus x squared and then plus C. And because 1/6 is a constant, then it will be absorbed by sea. And the final form will be x rays through the third power times, tangent inverse of X over three minus X squared over six plus 1/6 Ellen of one plus X squared, and then plus C.

To integrate this, we first want to apply substitution. And here you want to let w equal to X squared. And if we take the differential of both sides of this equation we get dw equal to two x dx or one half dw is just equal to X. Dx. Now since this integral is equal to X squared times erase two X squared times X and dx. Then this is just equal to the integral of W times E. Raised to W Times 1/2 dw. Or that's the same as 1/2 times the integral of W times E. Race to W. D. W. Now in here we will apply integration by parts. And you want to let U equal to W. And DV equal to the race to W. D. W. So the differential of you will be equal to D. W. And we will equal to erase to W. So from here we have one half times U. Times V. That will be W times erase to w minus the integral of V times D. You. That's the race to W. D. W. And then simplifying this, we have one half times W. Times the race to W minus E. Raised to W. And then plus C. and distributing one half. We get one half times W times erase to W minus one half times erase the W. And then plus C. Now since W is just X square, then this will be 1/2 times x squared times erased, two X squared minus one half times erased, X squared. And then plus C.

The problem is, first make a substitution and then use integration parts. You want to go into your O. R Signed Ellen Ax ax yet with this problem First beacon light. Why is they want you? You know, in next, use this substitution we have e y is equal to one over x and this integral asleep too into girl. Oh, look, Teo Park sign. Why? Why now? We can use integration. My past You wanted this into girl Fumbler is integral. You have to be from yaks if you come too. New tax lien minus into bro. You're primetime sleeve? Yes. Now for our problem, we can lie. T yu is cultural. Och, sign. Why on will be prom? His secret. Why then new prom one over one, minus y square on the BZ. To why so this into a girl utensil. Liza wine times cock sign. Why minus in general Your prom times he says this is why over. I want my sly square. You know why now for this part we can use u substitution game light. Hey, they cut you one month's. Why square then If you seek alternative to why You know why, Zico, Why not find. Why last long into grow. Hey, Over. Yeah. This isn't wine. Fine. Why us? This is T or we can What? It's a tough, for instance, cities, one slice choirs. Those This is one minus. Why square and don't forgot us passed into number C and then this wise it could tow you in next. This's a next hams ox sign. Nice mass returns one minus. Now our necks square on us. Constant number suit.

Again this question we have to do the integration of xQ sign x square D X. Using the table of integral. Ok. So after going through the table of intra girls, we cannot find a direct formula for these terms. Okay, so first of all we will transform it. First of all we will transform the integration and then we will apply the formula from develop integral. Okay, so uh we will substitute X. Inquiry calls to you. Okay then two x dx will become do you okay? Or we can say x dx will be do you divide it by two? Okay, so here it is Xq Synnex square that can be recognized X squared dot X. Okay. And sign X square and dx Now we will put the values there. Okay so this excess square will become you and X dx That will become do you divided by two and sign X square? It will be sign you okay? Sign you and you divided by Okay. And now it will be won by two. And integration of you sign you and do you? Okay? And now uh you sign you and the formula of this expression is available in our developed integral. Okay. So we were going to go through the Part three and Formula # 15. Okay. And I'm writing down the formula first that is integration of PX saying X. DX is equal to -1 by a P X. Because X and plus one by a square. P dash X. Sign X. Sign A X plus and so on. Plus constant in the last. Okay so we will apply this formula and do this. So while comparing we can say our function P X. Is you and the value of A is one here. Okay. And we have a variable you instead of X. So we will uh do this for our question. That is half integration of you sign you do you So it will be half already there and now we will do the integration part that is -1 by a. Okay so it will be -1x1. Okay A is one so it will be minus one by one and P X cost A X. So it will be you and cause you okay and plus it will be one by one square and prudish X. That is the differentiation of you, It will be one. Okay dot sign A X. That is signed. You where? Okay And plus constancy in the next time when we differentiate one it will become zero. So this will be the last time. Okay and now have you sign you okay when we uh it will be half integration of you sign you do you? It will be one x 2. Okay what we can say -1 by to you because you and it will be plus one by to sign you. Okay? And plus C. Constant. Okay. And now we will back substitute the value of you. That is access square. So left hand side when we back substitute. This will be our question that is xq sign X squared dx OK. And the right hand side it will be minus half you. Cause you So it will be X square and cause X square plus half. I knew then it will be sign makes a square and plus constant C. Okay so this will be the answer of our question. Thank you.


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