how do you determine the int eˆx(1-coseˆx) dx

Expert Answers

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You need to open the brackets such that:

`e^x*(1-cos e^x) = e^x - e^x*cos e^x`

Integrating both sides yields:

`int e^x*(1-cos e^x) dx= int e^x dx- int e^x*cos e^x dx`

`int e^x*(1-cos e^x) dx = e^x - int e^x*cos e^x dx`

You need to solve the integral `int e^x*cos e^x dx` , hence you should come up with the notation `e^x = y =gt e^x dx = dy`

You need to write `int e^x*cos e^x dx`  in terms of y such that:

`int e^x*cos e^x dx = int cos ydy = sin y + c`

You need to substitute`e^x`  for y such that:

`int e^x*cos e^x dx = sin e^x +`  c

`int e^x*(1-cos e^x) dx = e^x - sin e^x + c`

Hence, evaluating the integral of function `e^x*(1-cos e^x) ` yields `int e^x*(1-cos e^x) dx = e^x - sin e^x + c` .

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