Question

. Given f(x) = e 2x e 2x + 3e x + 2 : (a) Make the substitution u = e x to convert Z f(x) dx into an integral in u (HINT: The easiest way to do this in Python is to substitute into f(x) u 0 (x) ) (b) Your result from part (a) should be a rational function in u. Find the partial fraction decomposition (directly in Python) and use it to integrate the function. (Remember to substitute x back in when done!) (c) Evaluate Z f(x) dx directly.

Answer 1

Explanation:

Given;

`f(x)=(e^(2x))/(e^(2x)+3e^x+2)`

a)

substitute
`u=e^x\\du=e^x dx\\\\\int(e^(2x))/(e^(2x)+3e^x+2)dx=\int(e^x\dot e^x)/(e^x^(2x)+3e^x+2)dx\\\\=\int(udu)/(u^2+3u+2)`

b)

Apply partial fraction in (a), we get;

`(u)/(u^2+3u+2)=(2)/(u+2)-(1)/(u+1)\\\\\therefore u^2+3u+2\\=u^2+2u+u+2\\=u(u+2)+1(u+1)\\=(u+2)(u+1)\\\\Now\,\int(u)/(u^2+3u+2)\,du=\int(2)/(u+2)du-\int(1)/(u+1)du\\\\=2ln|u+2|-ln|u+1|+c\\=2ln|e^x+2|-lm|e^x+1|+c`

where C is an arbitrary constant