Integration as the Reverse of Differentiation

Mathematics · Calculus & Analysis · ages 17-18

Name ______________________ Date ____________

Why does a constant appear at the end of every one of these integrals?
- because a curve always meets the y-axis somewhere
- because dividing by the power leaves a remainder
- because the constant records where the curve crosses the y-axis, and the derivative cannot see the y-axis
- because infinitely many curves share that derivative, and it cannot say which one
Integrate 6x. What is the coefficient in front of the x² in your answer?
Answer: ______________
Differentiating your integral should give back something different from what you started with.
Circle one: True False
What is the integral of x³?
- 3x² + c
- x⁴ + c
- x⁴/4 + c
- x²/2 + c
What is the integral of 3x² + 8x?
- 6x + 8 + c
- x³ + 4x² + c
- 3x³ + 8x² + c
- x³ + 8x² + c
Someone claims the integral of 9x² + 4 is 3x³ + 4x + c. How do you check that without redoing the integration?
- substitute x = 1 into both expressions
- differentiate the answer and see whether it gives 9x² + 4
- add the two expressions and see whether they cancel
- set the answer equal to zero and solve for x
What is the integral of 12x²?
- 24x + c
- 6x³ + c
- 12x³ + c
- 4x³ + c
A curve has dy/dx = 6x² and passes through (1, 4). What is the curve?
- y = 2x³ + 4
- y = 6x³ + 2
- y = 2x³ + 2
- y = 2x³
A curve has dy/dx = 6x - 4 and passes through (2, 7). What is y when x = 1?
Answer: ______________
Ana integrates 8x³ and writes 2x⁴. Her powers and coefficients are right. What is still wrong?
- the + c is missing, so she has named one curve out of infinitely many
- the coefficient should have been 8 divided by 3
- the power should have gone down to 2, not up to 4
- nothing is wrong, since the constant can be assumed to be zero

Answer key

For grown-ups. Fold this page away before handing over the rest.

Integration as the Reverse of Differentiation W1-mt_edtk3ArxRk-s1

because infinitely many curves share that derivative, and it cannot say which one · x² + 1 and x² - 30 have the same derivative, and so do infinitely many others. Working backwards from 2x, nothing tells you which one was there, so + c stands for all of them.
3 · 6x is 6x¹, so the power goes to 2 and you divide the 6 by 2, giving 3.
False · Integration is the reverse of differentiation, so differentiating the answer has to return the original expression. That is exactly what makes the check work.
x⁴/4 + c · Raise the power from 3 to 4, then divide by 4. So the answer is x⁴/4 + c.
x³ + 4x² + c · 3x² raises to x³ and divides by 3, giving x³. Then 8x raises to x² and divides by 2, giving 4x². So the answer is x³ + 4x² + c.
differentiate the answer and see whether it gives 9x² + 4 · Integration and differentiation undo each other, so differentiating the proposed answer must return the original expression. Here 3x³ gives 9x² and 4x gives 4, so it checks out.
4x³ + c · The power goes from 2 to 3, and 12 is divided by the new power 3, giving 4x³ + c.
y = 2x³ + 2 · Integrating gives y = 2x³ + c. At (1, 4) that is 4 = 2 + c, so c = 2 and the curve is y = 2x³ + 2.
2 · Integrating gives y = 3x² - 4x + c. At (2, 7): 7 = 12 - 8 + c, so c = 3. Then at x = 1, y = 3 - 4 + 3 = 2.
the + c is missing, so she has named one curve out of infinitely many · 2x⁴ + 1 and 2x⁴ - 15 both differentiate to 8x³ as well. Without the + c the answer claims one particular curve when the question cannot single one out.