Rename the inside to undo the chain rule

Mathematics · Calculus & Analysis · ages 19-20

Name ______________________ Date ____________

Using radians and u = 4x+3, how does 8cos(4x+3) dx transform?
- 8cos(u) du
- 2cos(u) du
- cos(u) du/2
- 4cos(u) du
Using radians, which substitution changes 2x cos(x²) dx directly into cos(u) du?
- u = cos(x²)
- u = x²
- u = 2x
What is the indefinite integral of x e^(x²) dx?
- e^(x²)+C
- x e^(x²)+C
- e^(x²)/2+C
If u = x²+4, what is du?
- du = 2x dx
- du = 1 dx
- du = x² dx
For x bounds 0 to 3 and substitution u = 2-x, what is the corresponding ordered pair of u bounds?
- 0 to 3
- -1 to 2
- 2 to -1
- 2 to 3
Evaluate the integral from 0 to 1 of 3x²(1+x³) dx. Give an exact decimal.
Answer: ______________
Using radians, someone proposes sin(x²) as an antiderivative of cos(x²). What does differentiating the proposal reveal?
- Its derivative equals cos(x²) for every input.
- Its derivative has an extra multiplier 2x.
- Its derivative has the cosine argument replaced by 2x.
- Its derivative is identically zero for positive inputs.
Which substitution transforms x/(1+x²) dx specifically into du/(2u)?
- u = x
- u = x²
- u = 1/(1+x²)
- u = 1+x²
On x > 0, why does u = x² not reduce e^(x²) dx directly to the basic exponential differential e^u du?
- Exponentials cannot be used in any change of variables.
- Indefinite integrals require converted numerical bounds.
- The differential leaves an extra factor.
Evaluate the integral from 1 to e of (ln x)³/x dx. Give an exact decimal.
Answer: ______________

Answer key

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

2cos(u) du · Since du = 4 dx, the factor 8 dx becomes 2 du.
u = x² · With u = x², du = 2x dx matches the differential factor.
e^(x²)/2+C · The substitution u = x² introduces a factor 1/2.
du = 2x dx · Differentiate x²+4 to obtain 2x.
2 to -1 · The substitution is decreasing: x = 0 gives u = 2 and x = 3 gives u = -1.
1.5 · With u = 1+x³, the bounds are 1 and 2 and the integral of u is 3/2.
Its derivative has an extra multiplier 2x. · The chain rule gives 2x cos(x²), not the proposed integrand.
u = 1+x² · Only the supplied choice u = 1+x² makes the denominator u and x dx equal du/2.
The differential leaves an extra factor. · The valid change gives e^u du/(2sqrt(u)), so the direct basic exponential shortcut does not apply.
0.25 · With u = ln x, the integral becomes integral from 0 to 1 of u³ du, equal to 1/4.