Question (use tabular method
Studdy Solution
STEP 1
1. The integral requires integration by parts.
2. The tabular method is a systematic way to apply integration by parts repeatedly.
STEP 2
1. Identify functions for integration by parts.
2. Set up the tabular method.
3. Apply the tabular method to find the integral.
STEP 3
Identify the parts for integration by parts. The formula for integration by parts is:
Choose:
- (which will be differentiated)
- (which will be integrated)
STEP 4
Set up the tabular method. Create a table with two columns: one for derivatives of and one for integrals of .
STEP 5
Apply the tabular method. Multiply diagonally and alternate signs:
\begin{align*}
&+ (y^3)(\sin y) \\
&- (3y^2)(-\cos y) \\
&+ (6y)(-\sin y) \\
&- (6)(\cos y) \\
\end{align*}
So, the integral becomes:
where is the constant of integration.
The solution to the integral is:
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