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

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

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

Expert Answers

justaguide

| Certified 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

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Student Comments

giorgiana1976 | Student

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