What is the antiderivative of f(x)=x*e^4x.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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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