Calculus Worksheets | Definite Integration Worksheets - Free Printable
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Step-by-step solution for: Calculus Worksheets | Definite Integration Worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Calculus Worksheets | Definite Integration Worksheets
Let’s solve each problem using the Second Fundamental Theorem of Calculus.
This theorem says:
If \( F(x) = \int_{a}^{g(x)} f(t)\, dt \), then
\( F'(x) = f(g(x)) \cdot g'(x) \)
In simple words:
→ Plug the top limit (the function of x) into the integrand (replace t with that function).
→ Then multiply by the derivative of the top limit.
→ If the bottom limit is a constant, we don’t need to worry about it.
→ If the top limit is just “x”, then its derivative is 1, so you just plug in x.
We’ll go one by one.
---
Problem 1:
\( F(x) = \int_{6}^{2x} (-4t^2 - 5t)\, dt \)
Top limit: \( 2x \) → derivative = 2
Plug \( 2x \) into the integrand:
\( -4(2x)^2 - 5(2x) = -4(4x^2) - 10x = -16x^2 - 10x \)
Multiply by derivative of top limit (which is 2):
\( F'(x) = (-16x^2 - 10x) \cdot 2 = -32x^2 - 20x \)
✔ Final Answer for #1: \( -32x^2 - 20x \)
---
Problem 2:
\( F(x) = \int_{-10}^{2x^2} (7t^2 - 3t - 6)\, dt \)
Top limit: \( 2x^2 \) → derivative = 4x
Plug \( 2x^2 \) into integrand:
\( 7(2x^2)^2 - 3(2x^2) - 6 = 7(4x^4) - 6x^2 - 6 = 28x^4 - 6x^2 - 6 \)
Multiply by 4x:
\( F'(x) = (28x^4 - 6x^2 - 6) \cdot 4x = 112x^5 - 24x^3 - 24x \)
✔ Final Answer for #2: \( 112x^5 - 24x^3 - 24x \)
---
Problem 3:
\( F(x) = \int_{8}^{3x} (6t^2 - 7t + 5)\, dt \)
Top limit: \( 3x \) → derivative = 3
Plug \( 3x \) into integrand:
\( 6(3x)^2 - 7(3x) + 5 = 6(9x^2) - 21x + 5 = 54x^2 - 21x + 5 \)
Multiply by 3:
\( F'(x) = (54x^2 - 21x + 5) \cdot 3 = 162x^2 - 63x + 15 \)
✔ Final Answer for #3: \( 162x^2 - 63x + 15 \)
---
Problem 4:
\( F(x) = \int_{3}^{x} (4t^2 - 6t - 6)\, dt \)
Top limit: \( x \) → derivative = 1
Plug \( x \) into integrand:
\( 4x^2 - 6x - 6 \)
Multiply by 1 → same thing.
✔ Final Answer for #4: \( 4x^2 - 6x - 6 \)
---
Problem 5:
\( F(x) = \int_{5}^{3x^2} (-7t + 1)\, dt \)
Top limit: \( 3x^2 \) → derivative = 6x
Plug \( 3x^2 \) into integrand:
\( -7(3x^2) + 1 = -21x^2 + 1 \)
Multiply by 6x:
\( F'(x) = (-21x^2 + 1) \cdot 6x = -126x^3 + 6x \)
✔ Final Answer for #5: \( -126x^3 + 6x \)
---
Problem 6:
\( F(x) = \int_{-5}^{x} (6t^3 - 3t^2 - 3t + 1)\, dt \)
Top limit: \( x \) → derivative = 1
Plug \( x \) into integrand:
\( 6x^3 - 3x^2 - 3x + 1 \)
Multiply by 1 → same.
✔ Final Answer for #6: \( 6x^3 - 3x^2 - 3x + 1 \)
---
Problem 7:
\( F(x) = \int_{-6}^{3x^2} (-3t - 1)\, dt \)
Top limit: \( 3x^2 \) → derivative = 6x
Plug \( 3x^2 \) into integrand:
\( -3(3x^2) - 1 = -9x^2 - 1 \)
Multiply by 6x:
\( F'(x) = (-9x^2 - 1) \cdot 6x = -54x^3 - 6x \)
✔ Final Answer for #7: \( -54x^3 - 6x \)
---
Problem 8:
\( F(x) = \int_{4}^{x} (2t - 5)\, dt \)
Top limit: \( x \) → derivative = 1
Plug \( x \) into integrand:
\( 2x - 5 \)
Multiply by 1 → same.
✔ Final Answer for #8: \( 2x - 5 \)
---
Problem 9:
\( F(x) = \int_{10}^{x} (7t + 6)\, dt \)
Top limit: \( x \) → derivative = 1
Plug \( x \) into integrand:
\( 7x + 6 \)
Multiply by 1 → same.
✔ Final Answer for #9: \( 7x + 6 \)
---
Problem 10:
\( F(x) = \int_{0}^{x} (t^4 - 4t^2 - 7t - 4)\, dt \)
Top limit: \( x \) → derivative = 1
Plug \( x \) into integrand:
\( x^4 - 4x^2 - 7x - 4 \)
Multiply by 1 → same.
✔ Final Answer for #10: \( x^4 - 4x^2 - 7x - 4 \)
---
Final Answer:
1) \( -32x^2 - 20x \)
2) \( 112x^5 - 24x^3 - 24x \)
3) \( 162x^2 - 63x + 15 \)
4) \( 4x^2 - 6x - 6 \)
5) \( -126x^3 + 6x \)
6) \( 6x^3 - 3x^2 - 3x + 1 \)
7) \( -54x^3 - 6x \)
8) \( 2x - 5 \)
9) \( 7x + 6 \)
10) \( x^4 - 4x^2 - 7x - 4 \)
This theorem says:
If \( F(x) = \int_{a}^{g(x)} f(t)\, dt \), then
\( F'(x) = f(g(x)) \cdot g'(x) \)
In simple words:
→ Plug the top limit (the function of x) into the integrand (replace t with that function).
→ Then multiply by the derivative of the top limit.
→ If the bottom limit is a constant, we don’t need to worry about it.
→ If the top limit is just “x”, then its derivative is 1, so you just plug in x.
We’ll go one by one.
---
Problem 1:
\( F(x) = \int_{6}^{2x} (-4t^2 - 5t)\, dt \)
Top limit: \( 2x \) → derivative = 2
Plug \( 2x \) into the integrand:
\( -4(2x)^2 - 5(2x) = -4(4x^2) - 10x = -16x^2 - 10x \)
Multiply by derivative of top limit (which is 2):
\( F'(x) = (-16x^2 - 10x) \cdot 2 = -32x^2 - 20x \)
✔ Final Answer for #1: \( -32x^2 - 20x \)
---
Problem 2:
\( F(x) = \int_{-10}^{2x^2} (7t^2 - 3t - 6)\, dt \)
Top limit: \( 2x^2 \) → derivative = 4x
Plug \( 2x^2 \) into integrand:
\( 7(2x^2)^2 - 3(2x^2) - 6 = 7(4x^4) - 6x^2 - 6 = 28x^4 - 6x^2 - 6 \)
Multiply by 4x:
\( F'(x) = (28x^4 - 6x^2 - 6) \cdot 4x = 112x^5 - 24x^3 - 24x \)
✔ Final Answer for #2: \( 112x^5 - 24x^3 - 24x \)
---
Problem 3:
\( F(x) = \int_{8}^{3x} (6t^2 - 7t + 5)\, dt \)
Top limit: \( 3x \) → derivative = 3
Plug \( 3x \) into integrand:
\( 6(3x)^2 - 7(3x) + 5 = 6(9x^2) - 21x + 5 = 54x^2 - 21x + 5 \)
Multiply by 3:
\( F'(x) = (54x^2 - 21x + 5) \cdot 3 = 162x^2 - 63x + 15 \)
✔ Final Answer for #3: \( 162x^2 - 63x + 15 \)
---
Problem 4:
\( F(x) = \int_{3}^{x} (4t^2 - 6t - 6)\, dt \)
Top limit: \( x \) → derivative = 1
Plug \( x \) into integrand:
\( 4x^2 - 6x - 6 \)
Multiply by 1 → same thing.
✔ Final Answer for #4: \( 4x^2 - 6x - 6 \)
---
Problem 5:
\( F(x) = \int_{5}^{3x^2} (-7t + 1)\, dt \)
Top limit: \( 3x^2 \) → derivative = 6x
Plug \( 3x^2 \) into integrand:
\( -7(3x^2) + 1 = -21x^2 + 1 \)
Multiply by 6x:
\( F'(x) = (-21x^2 + 1) \cdot 6x = -126x^3 + 6x \)
✔ Final Answer for #5: \( -126x^3 + 6x \)
---
Problem 6:
\( F(x) = \int_{-5}^{x} (6t^3 - 3t^2 - 3t + 1)\, dt \)
Top limit: \( x \) → derivative = 1
Plug \( x \) into integrand:
\( 6x^3 - 3x^2 - 3x + 1 \)
Multiply by 1 → same.
✔ Final Answer for #6: \( 6x^3 - 3x^2 - 3x + 1 \)
---
Problem 7:
\( F(x) = \int_{-6}^{3x^2} (-3t - 1)\, dt \)
Top limit: \( 3x^2 \) → derivative = 6x
Plug \( 3x^2 \) into integrand:
\( -3(3x^2) - 1 = -9x^2 - 1 \)
Multiply by 6x:
\( F'(x) = (-9x^2 - 1) \cdot 6x = -54x^3 - 6x \)
✔ Final Answer for #7: \( -54x^3 - 6x \)
---
Problem 8:
\( F(x) = \int_{4}^{x} (2t - 5)\, dt \)
Top limit: \( x \) → derivative = 1
Plug \( x \) into integrand:
\( 2x - 5 \)
Multiply by 1 → same.
✔ Final Answer for #8: \( 2x - 5 \)
---
Problem 9:
\( F(x) = \int_{10}^{x} (7t + 6)\, dt \)
Top limit: \( x \) → derivative = 1
Plug \( x \) into integrand:
\( 7x + 6 \)
Multiply by 1 → same.
✔ Final Answer for #9: \( 7x + 6 \)
---
Problem 10:
\( F(x) = \int_{0}^{x} (t^4 - 4t^2 - 7t - 4)\, dt \)
Top limit: \( x \) → derivative = 1
Plug \( x \) into integrand:
\( x^4 - 4x^2 - 7x - 4 \)
Multiply by 1 → same.
✔ Final Answer for #10: \( x^4 - 4x^2 - 7x - 4 \)
---
Final Answer:
1) \( -32x^2 - 20x \)
2) \( 112x^5 - 24x^3 - 24x \)
3) \( 162x^2 - 63x + 15 \)
4) \( 4x^2 - 6x - 6 \)
5) \( -126x^3 + 6x \)
6) \( 6x^3 - 3x^2 - 3x + 1 \)
7) \( -54x^3 - 6x \)
8) \( 2x - 5 \)
9) \( 7x + 6 \)
10) \( x^4 - 4x^2 - 7x - 4 \)
Parent Tip: Review the logic above to help your child master the concept of fundamental theorem of calculus worksheet.
