What is the antiderivative of f(x)=x*e^4x.
2 Answers
justaguide | College Teacher | (Level 2) Distinguished Educator
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
giorgiana1976 | College Teacher | (Level 3) Valedictorian
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