Examine if f is primitive of g? f=e^x(x^2+1) g=e^x(x+1)^2

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to test if `f(x)` is the primitive of `g(x)` , hence, you need to check if `f'(x) = g(x)` , such that:

`f'(x) = (e^x(x^2+1))'`

You need to differentiate the function with respect to `x` , using the product rule, such that:

`f'(x) = (e^x)'(x^2 + 1) + e^x*(x^2 + 1)'`

`f'(x) = e^x*(x^2+1) + e^x*(2x)`

Factoring out `e^x` yields:

`f'(x) = e^x*(x^2 + 2x + 1)`

You need to notice that the factor `x^2 + 2x + 1` represents the expansion of squared binomial `(x + 1)^2` , such that:

`f'(x) = e^x*(x + 1)^2`

Comparing the equation of `f'(x)` with the equation of `g(x)` yields that they coincide, hence, the function `f(x)` represents the primitive of `g(x)` .

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