Question (Simplify your answer.)
Studdy Solution
STEP 1
What is this asking?
We need to find the definite integral of a simple polynomial from to .
Watch out!
Don't forget to apply the limits of integration correctly after finding the indefinite integral.
Also, remember the power rule for integration!
STEP 2
1. Find the indefinite integral.
2. Evaluate the definite integral.
STEP 3
Alright, let's **break down** this integral piece by piece!
We've got .
Remember, the *power rule* for integration says , where is the constant of integration.
We'll add the constant at the end.
STEP 4
First, we'll tackle the term.
Using the power rule, we get .
STEP 5
Next up is the term. .
STEP 6
Finally, the constant term . .
STEP 7
Putting it all together, the indefinite integral is , where is our constant of integration.
Since we're doing a definite integral, we can drop the for now!
STEP 8
Now for the grand finale!
We need to evaluate our indefinite integral, , from to .
STEP 9
Let's plug in our **upper limit**, : .
STEP 10
Now, the **lower limit**, : .
STEP 11
To get our **final answer**, we subtract the result of the lower limit from the result of the upper limit: .
STEP 12
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