what is the integral of x*e^x?

Answer 1

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Unlike Differentiation, you cannot perform the operation of Integration on combinations of functions. Hence, in order to integrate such functions, we use a method called Integration by parts.

Here, we try to split the given function into 2 separate functions, taking one as 'u' and the other as 'dv'. Deriving a formula from the product rule of differentiation, we can evaluate the integral as,

d (uv) = u.dv - v.du
Integrating on both sides...
∫d (uv) = ∫u.dv - ∫v.du
uv = ∫u.dv - ∫v.du

∫u.dv = uv - ∫v.du
The above equation gives us the formula for Integration by parts.

The order of priority to take a given function as 'u' is in the following order, which can be remembered as a mnemonic ILATE
I - Inverse
L - Logarithmic
A - Algebraic
T - Trigonometric
E - Exponential

Now, let us integrate the function you have given...

∫x * e^x . dx

According to ILATE,
u = x (an algebraic function), dv = e^x . dx (an exponential function)

∫u.dv = uv - ∫v.du

∫x * e^x . dx = uv - ∫v.du --------------------- (i)

u = x
Differentiating...
=> du = 1 . dx = dx { d(x^n) = n * x^(n-1) }

dv = e^x . dx
Integrating...
=> ∫dv = ∫e^x . dx
=> v = e^x { ∫e^(ax+b).dx = e^(ax+b) / a }

Now substituting the values in (i)...

∫x * e^x . dx = uv - ∫v.du
=> ∫x * e^x . dx = x * e^x - ∫e^x . dx
=> ∫x * e^x . dx = x * e^x - e^x
=> ∫x * e^x . dx = e^x [x - 1]

Therefore,

∫x * e^x . dx = e^x [x - 1] + c

- where c is the constant of integration.

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Answer 2

Formula for integration by parts is:-
I = Int(udv/dx)dx = uv - Int (vdu/dx)dx
where u = x and dv/dx = e^x
du/dx = 1 and v = e^x
I = xe^x - Int(e^x) + c
I = xe^x - e^x + c
I = e^x(x - 1) + c
Unlike Preety i am not majoring in maths, but i would sure like to know how to get the integral sign she uses instead of having to write Int all the time!

Answer 3

Integral (x e^x ) dx

You have to use integration by parts to solve this.

Let u = x. dv = e^x dx
du = dx. v = e^x

Therefore

uv - Integral (v du)

x e^x - Integral (e^x) dx

x e^x - e^x + C

Answer 4

The crucial of e^x is e^x. it is one reason the consistent e is so particular, and maximum every physique is taught that common crucial in calculus. crucial(e^x-e)dx=crucial(e^x)dx-integr... crucial of e is e*x (e is a relentless, comparable as e*x^0, use the flexibility rule, a million*e*x^a million=e*x) so it fairly is e^x-e*x+c (c being a relentless on account that this wasn't a different crucial)

Answer 5

From the product rule of differentiation we can deduce a formula for integration by parts. Let u=x and dv/dx = e^x
int. u(dv/dx) dx = uv - int. v(du/dx) dx
then,
int. x.e^x dx = xe^x - int. e^x dx = e^x (x - 1) + c

Answer 6

(x-1)e^x

Answer 7

int(x*e^x)dx===>
es una integral por partes este es el método
f*g´-int(g*f´)
f=x --->f´=dx
g´=e^x -->g=e^x (la int de e^u=e^u)
sust valores
x*e^x- int(e^x dx)= x*e^x-e^x+C

Answer 8

suppose I(x) is integral sign!
from the formula e^x=(e^x)'
I(x*e^xdx)=
=I(x*(e^x)'*dx)=
=x*e^x-I(x'*e^x*dx)=
=x*e^x-I(e^x*dx)=
=x*e^x-e^x
=e^x(x-1)