Efficient Integration Techniques: Expert Help for (e^x - 2)^2 Equation

[pnazari]

4/(e^x - 2)^2

That was a test question that I had tonight...solutions won't be up for a while, wondering how to approach it.

I thought of expanding the bottom bracket, then multiplying by (e^x)/(e^x), substituting u=e^x and then using partial fractions...not sure if it is right, thanks in advance.

 

[dextercioby]

Yes,i would have done it the same way:multiply both the numerator & the denominator with the e^{x} and then make the obvious substitution e^{x}---->u and then use partial fractions.

Daniel.

 

[wais]

Hello, thank you for reaching out for expert help with your integration question. Your approach of expanding the bottom bracket and using partial fractions is a good start. Let me walk you through the steps to solve this efficiently.

First, let's expand the bottom bracket to get 4/(e^x - 2)^2 = 4/(e^2x - 4e^x + 4). Then, we can substitute u = e^x to get 4/(u^2 - 4u + 4). Next, we can use partial fractions to split this into two separate fractions: 4/((u-2)^2) = A/(u-2) + B/(u-2)^2.

To solve for A and B, we can equate the numerators and simplify to get A(u-2) + B = 4. Plugging in u = 2, we get B = 4. Then, plugging in u = 0, we get A = -4. So our final equation becomes -4/(u-2) + 4/(u-2)^2.

Now, we can substitute back in e^x for u to get -4/(e^x - 2) + 4/(e^x - 2)^2. This is the final solution for our integral.

Remember to always check your work by taking the derivative of your solution to make sure it matches the original equation. I hope this helps and good luck on your test!