Question Statement
Evaluate the definite integral:
∫−21(12x5−36)dx
Background and Explanation
This problem requires evaluating a definite integral of a polynomial function. You will need to apply the linearity property of integrals (allowing you to integrate term by term) and the power rule for integration to find the antiderivative before evaluating at the given bounds.
Solution
We solve this by separating the integral into two parts, finding the antiderivative of each term, and then applying the Fundamental Theorem of Calculus.
Step 1: Apply linearity to separate the integral.
∫−21(12x5−36)dx=12∫−21x5dx−36∫−211dx
Step 2: Integrate each term using the power rule ∫xndx=n+1xn+1.
=12[5+1x5+1]−21−36[x]−21
=12[6x6]−21−36[x]−21
Step 3: Simplify the coefficient in the first term (12÷6=2).
=2[x6]−21−36[x]−21
Step 4: Evaluate both antiderivatives at the upper limit (x=1) and lower limit (x=−2).
=2[(1)6−(−2)6]−36[1−(−2)]
Step 5: Calculate the values inside the brackets.
=2[1−64]−36[1+2]
=2(−63)−36(3)
Step 6: Complete the arithmetic.
=−126−108
=−234
Therefore, the value of the definite integral is −234.
- Linearity of Integration: ∫ab[cf(x)+dg(x)]dx=c∫abf(x)dx+d∫abg(x)dx
- Power Rule for Integration: ∫xndx=n+1xn+1+C (for n=−1)
- Fundamental Theorem of Calculus: ∫abf(x)dx=F(b)−F(a), where F(x) is the antiderivative of f(x)
- Definite Integral Evaluation: [F(x)]ab=F(b)−F(a)
Summary of Steps
- Separate the integral into two parts: 12∫x5dx and −36∫1dx
- Find antiderivatives: 12x5 becomes 2x6 (since 12/6=2) and −36 becomes −36x
- Evaluate 2x6 from −2 to 1: 2[(1)6−(−2)6]=2[1−64]=−126
- Evaluate −36x from −2 to 1: −36[1−(−2)]=−36(3)=−108
- Add the results: −126+(−108)=−234