Question

Use the reduction formulas in a table of integrals to evaluate the following integrals. $$\int x^{3} e^{2 x} d x$$

Short answer

Question: Evaluate the integral $$\int x^{3} e^{2 x} d x$$ using reduction formulas. Answer: The integral is evaluated as $$x^3 e^{2x} (\frac{1}{8} - \frac{3}{8} x + \frac{3}{4} x^2 - \frac{3}{8} x^3) + C$$ where \(C\) is the constant of integration.

Step-by-step solution

Identify the parts for integration by parts

First, we need to identify the two parts for integration by parts. We will take the reduction formula found in the table of integrals which states that: $$\int x^n e^{ax} dx = \frac{1}{a^n} x^n e^{ax} - \frac{n}{a} \int x^{n-1} e^{ax} dx$$ In our case, we have \(n = 3\), \(a = 2\), and \(x^{3} e^{2x}\). You will see that we need to apply the formula until the exponent of x decreases.

Apply the reduction formula to \(n = 3\)

Applying the reduction formula to \(\int x^{3} e^{2x} dx\), we get: $$\int x^{3} e^{2x} dx = \frac{1}{8} x^3 e^{2x} - \frac{3}{2} \int x^{2} e^{2x} dx$$

Apply the reduction formula to \(n = 2\)

Now apply the formula to the new integral \(\int x^{2} e^{2x} dx\): $$\int x^{2} e^{2x} dx = \frac{1}{4} x^2 e^{2x} - \frac{2}{2} \int x^1 e^{2x} dx$$

Apply the reduction formula to \(n = 1\)

Lastly, apply the formula to the simplest integral \(\int x^1 e^{2x} dx\): $$\int x^1 e^{2x} dx = \frac{1}{2} x e^{2x} - \frac{1}{2} \int e^{2x} dx$$

Evaluate remaining integral

Now evaluate the remaining integral \(\int e^{2x} dx\): $$\int e^{2x} dx = \frac{1}{2} e^{2x} + C$$

Substitute steps 4, 3, and 2 to find the final solution

Replace the simplified integral in step 4: $$\int x^1 e^{2x} dx = \frac{1}{2} x e^{2x} - \frac{1}{2} (\frac{1}{2} e^{2x} + C)$$ Now substitute this into step 3: $$\int x^2 e^{2x} dx = \frac{1}{4} x^2 e^{2x} - (\frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x}+ C_1)$$ Finally, substitute this into step 2: $$\int x^{3} e^{2 x} d x = \frac{1}{8} x^3 e^{2x} - \frac{3}{2} (\frac{1}{4} x^2 e^{2x} - (\frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x}+ C_1)) + C_2$$

Simplify the final solution

Now, simplify the expression above to obtain the final solution: $$\int x^{3} e^{2 x} d x = \frac{1}{8} x^3 e^{2x} - \frac{3}{8} x^2 e^{2x} + \frac{3}{4} xe^{2x} - \frac{3}{8} e^{2x} + C = x^3 e^{2x} (\frac{1}{8} - \frac{3}{8} x + \frac{3}{4} x^2 - \frac{3}{8} x^3) + C$$