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Question Details LarCalc11 8.R.039_Use partial fractions to find the indefinite integral. (Use C for the constant of integration_ Remember to use absolute values w...

Question

Question Details LarCalc11 8.R.039_Use partial fractions to find the indefinite integral. (Use C for the constant of integration_ Remember to use absolute values where appropriate: )x2 + 6xf x _x2 +X- 1

Question Details LarCalc11 8.R.039_ Use partial fractions to find the indefinite integral. (Use C for the constant of integration_ Remember to use absolute values where appropriate: ) x2 + 6x f x _x2 +X- 1



Answers

Use partial fractions to find the indefinite integral. $$ \int \frac{1}{2 x^{2}-x} d x $$

In the problem we have been given integration to upon X into x minus two, the X. So here this can be written as two upon. This is too so this is to upon X and the x m minus two is equal to AARP, onyx plus be upon x minus two. Now here this is written as to that equals two, A into x minus two plus in the X fund them. After solving we found the value of A equal to minus one and the equal to wanna. So these are the values of A and B. Further we have integration minus one upon X plus one upon x minus two dx. So this is written as minus Ellen more X plus, Ellen mod x minus two plus C. All this is written as Ellen mod x minus two upon X plus C. This is the answer.

In the problem we have been given integration X plus two upon X into x minus four. The X this is written as X plus two upon x into x minus four is equal to a bon x plus be upon x minus four, hence it is equal to X plus two. That is equal to a into x minus four plus bx. Now we have this as to which is equal to a N do minus four or a equals minus one upon to six equal to for B r b equal to 3.2. Hence, further this is written as integration minus one upon two, X plus three upon to in two x minus four dx. This equals two minus one upon to a landmark X plus three upon to Ellen mode X minus four plus say Or it is equal to one upon to into three Ln mod x minus four minus Ln model X plus see. So this is the answer.

In the problem we have been given no aggression five that is five upon it's a Squire plus three x minus two X plus it is minus so minus six. Now this is written as integration five upon x minus two in two. X plus three The X. So this is written here as Fire born x minus two in two weeks plus three. This is written as A upon x minus two plus B upon X plus three. So further this is a fight that equals two. Eight weeks plus three plus being two x minus two. Hence we obtain the values of A. That equals one and B. That equals minus one. So this is integration one about x minus two minus one upon X plus three. The X. That is written as Ellen more x minus two minus integration sorry minus Ellen mud X plus three plus c authorities. Ellen mud X minus two upon X plus three plus C. So this is the answer.

For this problem we need to use partial fractions to find the indefinite integral. Now we can rewrite the immigrant into exports to over X squared plus five. X. Is equal to expose to over. We have x times x plus five and we write this into partial fractions A over X plus we have be over X plus five. Now, multiplying both sides of the equation by the L C. D. X times X plus five. We would have X plus two Equal to eight times expose five plus B times X. Now we want to find A. And B. So that we can complete our partial fractions. And to do that we um fix a value for X. Let's say X is equal to -5. And and here we get negative five plus two equal to eight times negative five plus five plus B. Times negative five. And this would give us -3 equal to negative five. B Or B is just equal to 3/5. And we want to fix another X value let's say X equals zero to find E. And in here we have zero plus two equal to eight times 0 plus five Plus B Times zero. This would give us two equal to five a. Or that A. is equal to 2/5. Now in choosing these X values, we simply set the denominator 20 and sulfur X. And this would give us the convenient access values that we used in the previous step. And so now that we have A. And B. Then we have X -2 over X squared plus five X. This is just 2/5 times one over X. plus we have 3/5 times one over X plus five. And so integrating both sides with respect to X. Then you would have 2/5 L. An absolute value of X plus we have 3/5. Ellen absolute value of X plus five and then plus C. And so by partial fractions. This is the indefinite integral


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