Evaluate \$ \int (x^2 * e^x) dx\$.

Let’s consider the integral as follows:

\$ \int (x^2 * e^x) dx\$

Using the technique of integration by parts, we choose

\$ u = x^2 and dv = e^x dx \$

This gives us

\$ du = 2x dx \$

\$ v = \int e^x dx = e^x \$

Applying the integration by parts formula:

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

After simplifying the integral on the right-hand side, our expression becomes:

\$ \int (x^2 * e^x) dx = x^2 * e^x - \int e^x * 2x dx \$
\$ \int (x^2 * e^x) dx = x^2 * e^x - 2 \int x * e^x dx \$

Let’s focus on the remaining integral. We can use integration by parts again

\$ \int x * e^x dx \$
\$ u = x and dv = e^x dx \$
\$ du = dx and v = e^x \$

After applying the integration by parts formula again, simplify the resulting equation

\$ \int u dv = uv - \int v du \$
\$ \int ( x * e^x) dx = x * e^x - \int e^x dx \$
\$ \int ( x * e^x) dx = x * e^x - e^x \$

If we take \$e^x\$ as a common factor, the equation becomes:

\$ \int ( x * e^x) dx = e^x(x - 1) \$

Substituting this result back into our original equation:

\$ \int (x^2 * e^x) dx = x^2 * e^x - 2x * e^x + 2 * e^x + C \$

So, the value of the integral \$ \int (x^2 * e^x) dx \$ is \$ x^2 * e^x - 2x * e^x + 2 * e^x + C \$, where C is the constant of integration.

Option A is incorrect as it subtracts \$2⋅e^x\$ instead of \$2x ⋅ e^x\$ and doesn’t integrate the term involving \$x^2\$ correctly

\$ x^2 * e^x - 2 * e^x + 2 * e^x + C \$

Option B is incorrect because it doesn’t integrate correctly and includes a constant term that shouldn’t be there in the integral

\$ x^2 * e^x - 2x * e^x + 2 + C \$

Option C is incorrect because it omits \$x^2 * e^x\$ in the integral and presents \$x^2\$ outside the integral, violating the integration-by-parts method

\$ x^2 - 2x * e^x + 2 * e^x + C \$

The correct derivative for the given function appears to have been accurately determined

\$ x^2 * e^x - 2x * e^x + 2 * e^x + C \$

Option

D

Can you summarize what you’ve understood in the above steps?