Find the integral of x e^x dx.

Find the integral of x e^x dx.

• A. e^x (x - 1) + C
• B. e^x (x + 1) + C
• C. e^x (x^2/2) + C
• D. e^x (x^2 - 1) + C

Answer: (A)

To find the integral of \( x e^x \, dx \), we can use the technique of integration by parts. This method is particularly useful when dealing with products of functions, like \( x \) and \( e^x \). We start by letting: - \( u = x \) (which we will differentiate) - \( dv = e^x \, dx \) (which we will integrate) Next, we need to compute \( du \) and \( v \): - Differentiating \( u \) gives us \( du = dx \). - Integrating \( dv \) yields \( v = e^x \). We then apply the integration by parts formula, which states: \[ \int u \, dv = uv - \int v \, du \] Substituting our choices into this formula, we have: \[ \int x e^x \, dx = x e^x - \int e^x \, dx \] Now, we need to evaluate the integral on the right: \[ \int e^x \, dx = e^x \] Substituting this back into our earlier equation, we have: \[ \

To find the integral of ( x e^x , dx ), we can use the technique of integration by parts. This method is particularly useful when dealing with products of functions, like ( x ) and ( e^x ).

We start by letting:

• ( u = x ) (which we will differentiate)

• ( dv = e^x , dx ) (which we will integrate)

Next, we need to compute ( du ) and ( v ):

• Differentiating ( u ) gives us ( du = dx ).

• Integrating ( dv ) yields ( v = e^x ).

We then apply the integration by parts formula, which states:

[

\int u , dv = uv - \int v , du

]

Substituting our choices into this formula, we have:

[

\int x e^x , dx = x e^x - \int e^x , dx

]

Now, we need to evaluate the integral on the right:

[

\int e^x , dx = e^x

]

Substituting this back into our earlier equation, we have:

[

\