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Demonstrate that F(x)=e^xsinx is primitive of f(x)=e^x(sinx+cosx)?

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ruals | Salutatorian

Posted September 27, 2013 at 5:54 PM via web

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Demonstrate that F(x)=e^xsinx is primitive of f(x)=e^x(sinx+cosx)?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted September 27, 2013 at 6:16 PM (Answer #1)

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

`F'(x) = (e^x*sin x)'`

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

`F'(x) = (e^x)'*sin x + e^x*(sin x)'`

`F'(x) = e^x*sin x + e^x*cos x`

Factoring out `e^x` yields:

`F'(x) = e^x*(sin x + cos x)`

Comparing the equation of F'(x) with the equation of `f(x)` yields that they coincide.

Hence, testing if `F(x)` is the primitive of `f(x)` yields that `F'(x)` `= f(x)` , hence, the statement holds.

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