Question

In Exercises \(1-10,\) find the indefinite integral. $$\int 3 x^{2} e^{2 x} d x$$

Short answer

The indefinite integral \(\int 3 x^{2} e^{2 x} d x = 1.5 x^{2} e^{2x} - 3x e^{2x} + 1.5 e^{2x} + C\).

Step-by-step solution

Identifying u and dv

Choose \(u\) to be the algebraic part and \(dv\) to be the exponential part of the given integral. In this example, we will let \(u = 3x^2\) and \(dv = e^{2x} dx\).

Calculating du and v

After we choose \(u\) and \(dv\), we have to find \(du\) by differentiating \(u\) and \(v\) by integrating \(dv\). We get \(du = 6x dx\) and \(0.5 e^{2x}\).

Applying Integration by Parts

Finally, we use integration by parts formula which gives us: \(\int u dv = uv - \int v du\). We substitute the \(u, dv, du, v\) we found into this formula. So we get \(\int 3 x^{2} e^{2 x} d x = 3x^2 * 0.5 e^{2x} - \int 0.5 e^{2x} * 6x dx\). We notice that the remaining integral is also a case for integration by parts.

Solving the Remaining Integral

We proceed with the remaining integral by choosing \(u=x\) and \(dv=6 e^{2x} dx\). Therefore, \(du=dx\) and \(v=3 e^{2x}\). After applying integration by parts and simplifying, we get \(-3x e^{2x} + 3\int e^{2x} dx = -3x e^{2x} + 1.5 e^{2x}\).

Final Solution

Including the expression from the first application of integration by parts, we find the indefinite integral \(\int 3 x^{2} e^{2 x} d x = 1.5 x^{2} e^{2x} - 3x e^{2x} + 1.5 e^{2x} + C\), with \(C\) being the constant of integration.