Problem 38 - Essential Calculus

Question

Use the tabular method to find the integral. $$ \int x^{3} \cos 2 x d x $$

Step-by-Step Solution

Answer

The integral of \(x^{3} \cos 2 x d x\) is \(0.5x^{3}\sin 2x + 0.75x^{2}\cos 2x - 0.75x\sin 2x - 0.375\cos 2x + C\).

Step 1: Set up the table

Create a table with two columns. Write \(x^{3}\) in the first row of the left column, and \(\cos 2x \) in the first row of the right column. The function to differentiate (\(x^3\)) goes in the left column, while the function to integrate (\(\cos 2x\)) goes in the right. Now, compute successive derivatives of \(x^{3}\) and integrals of \(\cos 2x\), filling out the rest of the rows until reaching a row where the derivative of \(x^{3}\) is zero.

Step 2: Compute Derivatives and Integrals

Calculate the derivatives and integrals row by row. The derivatives of \(x^{3}\) are:1st derivative: 3x^{2}2nd derivative: 6x3rd derivative: 64th derivative: 0The integrals of \(\cos 2x\) are:1st integral: \(0.5\sin 2x \)2nd integral: -0.25\cos 2x3rd integral: -0.125\sin 2x4th integral: 0.0625\cos 2x.

Step 3: Multiply Diagonal and Add

Now, multiply along diagonals from upper left to lower right and sum up these products to evaluate the integral. Additionally, remember to alternate signs: plus, then minus, then plus, etc. So, \(x^{3} * 0.5\sin 2x - 3x^{2} * -0.25\cos 2x + 6x * -0.125\sin 2x - 6 * 0.0625\cos 2x + C\)

Step 4: Simplify the Expression

Simplify the resulting expression to obtain the final result. So, \(0.5x^{3}\sin 2x + 0.75x^{2}\cos 2x - 0.75x\sin 2x - 0.375\cos 2x + C\)