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Chapter 6: Problem 38

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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, \(x^{3} * 0.5\sin 2x - 3x^{2} * -0.25\cos 2x + 6x * -0.125\sin 2x - 6 * 0.0625\cos 2x + C\)
04

Simplify the Expression

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

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Most popular questions from this chapter

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