Integrate the functions e^x x².

To integrate the function \( e^{x} \cdot x^{2} \), we can use the technique of integration by parts. This method is useful when dealing with the product of two functions.

Integration by Parts Formula

The formula for integration by parts is:

∫ u dv = uv - ∫ v du

Here, we need to choose \( u \) and \( dv \) from our function \( e^{x} \cdot x^{2} \).

Choosing u and dv

• Let: \( u = x^{2} \) (which simplifies when differentiated)
• Then: \( du = 2x \, dx \)
• Let: \( dv = e^{x} \, dx \)
• Then: \( v = e^{x} \)

Applying the Formula

Now, we can apply the integration by parts formula:

∫ \( e^{x} \cdot x^{2} \, dx = x^{2} \cdot e^{x} - ∫ e^{x} \cdot 2x \, dx \)

Integrating the Remaining Integral

Next, we need to integrate \( ∫ e^{x} \cdot 2x \, dx \) using integration by parts again:

• Let: \( u = 2x \)
• Then: \( du = 2 \, dx \)
• Let: \( dv = e^{x} \, dx \)
• Then: \( v = e^{x} \)

Applying the formula again gives:

∫ \( e^{x} \cdot 2x \, dx = 2x \cdot e^{x} - ∫ 2e^{x} \, dx \)

Final Steps

The integral of \( 2e^{x} \) is simply \( 2e^{x} \). So we have:

∫ \( e^{x} \cdot 2x \, dx = 2x \cdot e^{x} - 2e^{x} \)

Putting It All Together

Now, substituting back into our original integration:

∫ \( e^{x} \cdot x^{2} \, dx = x^{2} \cdot e^{x} - (2x \cdot e^{x} - 2e^{x}) + C \)

Thus, the final result is:

∫ e^{x} \cdot x^{2} \, dx = e^{x}(x^{2} - 2x + 2) + C