Question

State whether you would use integration by parts to evaluate the integral. If so, identify what you would use for \(u\) and \(d v\). Explain your reasoning. $$ \int x^{2} e^{2 x} d x $$

Short answer

Yes, the integral \(\int x^{2} e^{2x} dx\) can be evaluated using integration by parts. For the first application of integration by parts, one would use \(u = x^2\) and \(dv = e^{2x} dx\). The integral can be simplified to \(0.5x^2e^{2x} - \int x e^{2x} dx\). The process needs to be repeated for the resulting integral \(\int x e^{2x} dx\), by taking \(u = x\) and \(dv = e^{2x} dx\).

Step-by-step solution

Identify u and dv

Choose \(u\) and \(dv\) for integration by parts. A rule of thumb is to choose \(u\) as a function that becomes simpler when differentiated, and \(dv\) as a function that doesn't become more complicated when integrated. So, it's suggested here to take \(u = x^2\) and \(dv = e^{2x} dx\).

Find du and v

After identifying \(u\) and \(dv\), compute \(du\) and \(v\). Here, the differential of \(u\), \( du = 2x dx\), is derived by differentiating \(u\). The function \(v\), \(v = 0.5e^{2x}\), is obtained by integrating \(dv\).

Apply the integration by parts formula

After identifying \(u\), \(v\), \(du\), and \(dv\), input them into the formula \(\int u dv = uv - \int v du\). Substituting the known values, we get \(\int x^{2} e^{2x} dx = x^2 * (0.5e^{2x}) - \int 0.5e^{2x} * 2x dx\). This simplifies to \(0.5x^2e^{2x} - \int x e^{2x} dx\).

Repeat the process

Notice that the new integral \(\int x e^{2x} dx\) is still a product of a polynomial in \(x\) and an exponential function, so it can again be handled with the method of integration by parts. Again, let \(u = x\), \(dv = e^{2x} dx\), calculate \(du = dx\) and \(v = 0.5e^{2x}\), and apply the formula for integration by parts. The final answer will be reached after this step.