Question (use tabular method
Studdy Solution
STEP 1
1. We are dealing with an indefinite integral.
2. The tabular method (also known as integration by parts) is to be used.
3. The basic formula for integration by parts is: .
STEP 2
1. Set up the tabular method.
2. Fill in the table with derivatives and integrals.
3. Apply the tabular method formula.
4. Simplify and combine terms.
STEP 3
Let's set up the tabular method. We'll create a table with two columns: one for derivatives of and one for integrals of .
\begin{array}{|c|c|}
\hline
\text{Derivatives of } y^3 & \text{Integrals of } \cos y \\
\hline
y^3 & \sin y \\
\hline
3y^2 & -\cos y \\
\hline
6y & -\sin y \\
\hline
6 & \cos y \\
\hline
0 & \sin y \\
\hline
\end{array}
STEP 4
Now, let's explain how we filled in the table:
1. In the left column, we start with and take derivatives until we reach 0.
2. In the right column, we start with and integrate repeatedly.
3. The signs in the right column alternate because the derivative of is , but the derivative of is .
STEP 5
To apply the tabular method, we use the following pattern:
1. Multiply entries diagonally, starting from the top left.
2. Alternate between adding and subtracting these products.
Let's apply this:
\begin{align*}
\int y^3 \cos y \, dy &= y^3 \sin y - 3y^2(-\cos y) - 6y(-\sin y) - 6(\cos y) + C \\
&= y^3 \sin y + 3y^2 \cos y + 6y \sin y - 6 \cos y + C
\end{align*}
STEP 6
The expression is already in its simplest form. We can write our final answer as:
Where is the constant of integration.
The solution to the indefinite integral using the tabular method is:
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