Find the integral of x^n e^x dx using integration by parts. What is the general form?

Find the integral of x^n e^x dx using integration by parts. What is the general form?

• A. (x^n e^x - n∫ x^(n-1) e^x dx) + C
• B. (x^n e^x + n∫ x^(n-1) e^x dx) + C
• C. (x^n e^x + n∫ x^(n+1) e^x dx) + C
• D. (x^n e^x - n∫ x^n e^x dx) + C

Answer: (A)

To find the integral of \( x^n e^x \, dx \) using integration by parts, we first choose our functions to differentiate and integrate. We typically set \( u = x^n \) (which we will differentiate) and \( dv = e^x \, dx \) (which we will integrate). From this choice, we have: - \( du = n x^{n-1} \, dx \) - \( v = e^x \) Now applying the integration by parts formula, which states \( \int u \, dv = u v - \int v \, du \), we can substitute our values: \[ \int x^n e^x \, dx = x^n e^x - \int e^x (n x^{n-1}) \, dx \] This simplifies to: \[ \int x^n e^x \, dx = x^n e^x - n \int x^{n-1} e^x \, dx \] Rearranging gives us the expression for the integral: \[ \int x^n e^x \, dx = x^n e^x -

To find the integral of ( x^n e^x , dx ) using integration by parts, we first choose our functions to differentiate and integrate. We typically set ( u = x^n ) (which we will differentiate) and ( dv = e^x , dx ) (which we will integrate).

From this choice, we have:

• ( du = n x^{n-1} , dx )

• ( v = e^x )

Now applying the integration by parts formula, which states ( \int u , dv = u v - \int v , du ), we can substitute our values:

[

\int x^n e^x , dx = x^n e^x - \int e^x (n x^{n-1}) , dx

]

This simplifies to:

[

\int x^n e^x , dx = x^n e^x - n \int x^{n-1} e^x , dx

]

Rearranging gives us the expression for the integral:

[

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