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Problem 4. (15 points) Use the integration by parts to evaluate the indefinite integral. fctan (x) dx...

Question

Problem 4. (15 points) Use the integration by parts to evaluate the indefinite integral. fctan (x) dx

Problem 4. (15 points) Use the integration by parts to evaluate the indefinite integral. fctan (x) dx



Answers

Evaluate the given indefinite integral. $$ \int\left(15 x^{-1}-4 \sinh x\right) d x $$

Okay. In this question there the interrogation of X cubed divided by X rays to the power four minus five Holy Square and D X. Okay. And in this question, we have to solve mhm then triggered by making the given substitution. And the substitution given in the question is you equals two X rays to the power four minus five. Okay, so we will do this. Substitution U equals to express to the power four minus five that will give us d u equals to four x cubed dx. OK. And the value of X cubed DX will be Do you divided by four and we can see X cubed dx is presenting our numerator. Okay, then X cubed, divided by x rays to the bar four minus five squared dx will be X cubed dx Well, with the U divided by four that is one by four hair and access to the about four minus five will be you. Then it will be us choir. Okay. Or we can say one by four. Interrogation You raised to the power minus to be you. Okay. And the integration of it one by four and integration of us raised to the power minus two. It will be minus one upon you plus See Okay. And now we will back. Substitute the value of you that is expressed to the power four minus five And it will be won by four and minus one. Divided by X rays to the power four minus five. Let's see. Okay. Or we can say it will be simple form minus one, divided by four x rays to the about four minus five plus c And that will be our final answer. Thank you.

Let's evaluate the integral of X to the fourth over X minus one. Here, the first thing we noticed is that the numerator has larger than read. So let's go out in these long division X minus one going into X to the fourth power we have executed. We'Ll supply that subtract left over with X cubed so we can add X squared up here we had X cubed minus X squared That's over with X squared so we could come back up here at an ex and then we'll have X squared minus x subtract left over with X And then we could come back here once more and then add that one in there and then we're X minus one Subtract that off and we had a one left over X Does not go into one. So this is our remainder. So we could come and write this as well. We just did the division our quotient was X cubed Ex player accident one and then we have our remainder divided by the question the original divisor X minus one and this is a much easier in the world. In the original, we can use the power rule for the first four terms for the last one, and may or may not help you to use the use up here. If you need to use because of the minus one, go ahead and do the U equals X minus one. So using the power rule and then the natural log, we have extra the fourth over. Four. Execute over three X squared over two X and then plus natural law. Don't forget that. Absolute value X minus one inside and closer constancy of integration and there's our answer.

Okay, We know the problems Tools we can substitute you is extra The fourth minus five, which means that we get d'you we take the derivative is four x cubed DD X, which gives us X cube de axe is do you divide by four Which means pulling out the constant Remember, Constable, the constant out of ventricles we have thus and we know that we we know the integral one over you is natural log of you pussy Sonata Final step substitute back in Obviously we don't want you in the final answer and we have our solution.

From the given integer We have to use you for 22 places to the powerful defensive in both sex would have to eat ideological took for a next cube DX. So we get 104. Do you just work? Excuse eggs So the indignant becomes one by four. Until this Enough You took about Why do you that is a photo one by four into U to the power six by six. Let's see that is a 4 to 1 by 24 persecuting you just two plus f to the power for the power six. Let's see.


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