Question

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

Short answer

The short answer is \(\frac{3}{2} t e^{2t} - \frac{3}{4} e^{2t} + C\).

Step-by-step solution

Identify the functions u and dv

Identify the two functions in your integral that will become u and dv in the formula for integration by parts. Here, \(u = 3t\) and \(dv = e^{2 t} dt\).

Derive du and integrate dv

Once we've identified u and dv, we should differentiate u to give us du and integrate dv to find v. Thus, \(du = 3 dt\) and \(v = \frac{1}{2} e^{2t}\).

Apply the integration by parts formula

Apply the formula for integration by parts: \(\int u dv = uv - \int v du\). This gives us the integral \(\int 3t e^{2t} dt = 3t \left(\frac{1}{2} e^{2t}\right) - \int \frac{1}{2} e^{2t} \times 3 dt\).

Solve the remaining integral

Now, we only have to solve the remaining integral \(\int \frac{3}{2} e^{2t} dt\). Taking out the constants, we get \(\frac{3}{2} \int e^{2t} dt = \frac{3}{2} \times \frac{1}{2} e^{2t} = \frac{3}{4} e^{2t}\).

Substitute back and simplify

Substitute the solved integral back into our equation from step 3, giving us \(3t \left(\frac{1}{2} e^{2t}\right) - \frac{3}{4} e^{2t} = \frac{3}{2} t e^{2t} - \frac{3}{4} e^{2t} + C\).