What is the antiderivative of f(x)=x*e^4x.
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We need to find the antiderivative of f(x)=x*e^4x
We solve this problem using Integration by parts:
Int [f(x)g'(x) dx = f(x)g(x)- Int [ f'(x)g(x) dx]
Let f(x) = x and g'(x)= e^4x
=> g(x) = e^4x/4
Int [ x*e^4x dx] = x*e^4x / 4 - Int [ e^4x/4 dx]
=> x* e^4x/4 - e^4x / 16 + C
Therefore the antiderivative of f(x)=x*e^4x is x*e^4x/4 - e^4x /16 + C
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To determine what is requested by enunciation, we'll have to evaluate the indefinite integral of the given function.
Int f(x)dx
We'll integrate by parts, so, we'll recall the formula:
Int udv = u*v - Int vdu (*)
We'll put u = x. (1)
We'll differentiate both sides:
du = dx (2)
We'll put dv = e^4x (3)
We'll integrate both sides:
Int dv = Int e^4x dx
v = e^4x/4 (4)
We'll substitute (1) , (2) , (3) and (4) in (*):
Int udv = x*e^4x/4 - Int (e^4x/4)dx
The anti-derivative is:
Int (x*e^4x)dx = (x*e^4x)/4 - (e^4x)/16 + C
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