Need help with these questions on Integration

Brpaulo

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May 25, 2019
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Hi I need help to solve these problems. Not sure whether I can use Integration by Partial Fractions or Completing the Square to approach these questions. Any help would be much appreciated. Thanks.

integration assignment.PNG
 
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Suppose we see what happens when we complete the square on the denominator of the integrand in the first problem. What would that look like?
 
I suppose that would give the quadratic formula, but I don't know how to proceed after that. If I tried to substitute the whole formula into x in the question, that makes the problem even more complicated to solve.
 
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I suppose that would give the quadratic formula, but I don't know how to proceed after that. If I tried to substitute the whole formula into x in the question, that makes the problem even more complicated to solve.
Well, no, the quadratic formula tells what values of x make it zero. You wouldn't want to replace x with those constants.

Completing the square will change the expression to [MATH](x - h)^2 + k^2[/MATH] or [MATH](x - h)^2 - k^2[/MATH], where you can use h and k in place of the expressions you get in a, b, and c to keep things simple (until the end).

Do you know any way to integrate an expression with one of those forms in the denominator?
 
I suppose that would give the quadratic formula, but I don't know how to proceed after that. If I tried to substitute the whole formula into x in the question, that makes the problem even more complicated to solve.

Well, let's complete the square on that denominator:

[MATH]ax^2+bx+c=a\left(x^2+\frac{b}{a}+\left(\frac{b}{2a}\right)^2\right)+c-\frac{b^2}{4a}=\frac{1}{4a}\left(\left(2ax+b\right)^2+\left(\sqrt{4ac-b^2}\right)^2\right)=\frac{1}{4a}\left(\left(\frac{2ax+b}{\sqrt{4ac-b^2}}\right)^2+1\right)[/MATH]
Can you proceed?
 
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