If f' (x) = x43, then f(x) = (1/44) x44 + C.
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-e-x + C.
You can't, unless it's an initial value problem. If f(x) is an antiderivative to g(x), then so is f(x) + c, for any c at all.
One can use integration by parts to solve this. The answer is (x-1)e^x.
-1
If you mean 2x times -2x then it would be -8x.