If
𝑓
is continuous and
∫
1
243
𝑓
(
𝑡
)
𝑑
𝑡
=
59
,
evaluate the integral
∫
1
3
3
𝑡
4
𝑓
(
𝑡
5
)
𝑑
𝑡
.

Question

If 
𝑓
 is continuous and
∫
1
243
𝑓
(
𝑡
)
𝑑
𝑡
=
59
,
evaluate the integral
∫
1
3
3
𝑡
4
𝑓
(
𝑡
5
)
𝑑
𝑡
.

BlackTom AI · Accepted Answer

Use substitution u = t^5 because the integrand contains f(t^5) and a t^4 factor. This matches the change-of-variable formula for definite integrals.

1) Relevant knowledge points and concepts
- Change of variables (substitution) in definite integrals: if u = g(t) is differentiable and g′(t) appears (or can be arranged) in the integrand, then ∫ f(g(t)) g′(t) dt = ∫ f(u) du with corresponding limits.
- Continuity of f ensures f is integrable on [1,243], so the substitution is valid.

2) Step-by-step reasoning
Let u = t^5. Then du/dt = 5 t^4, so du = 5 t^4 dt and t^4 dt = du/5.

3) Intermediate algebra and change of limits
When t = 1, u = 1^5 = 1.
When t = 3, u = 3^5 = 243.

Now rewrite the integral:
∫_{1}^{3} 3 t^4 f(t^5) dt
= 3 ∫_{1}^{3} t^4 f(t^5) dt
= 3 ∫_{u=1}^{u=243} f(u) (du/5)
= (3/5) ∫_{1}^{243} f(u) du.

4) Use the given value and compute
Given ∫_{1}^{243} f(t) dt = 59, substitute:
(3/5) × 59 = 177/5.

5) Why the correct answer is correct and why other choices are incorrect
- Correct: 177/5 follows directly from the substitution factor 1/5 coming from du = 5 t^4 dt and the overall multiplier 3, yielding (3/5)*59.
- Incorrect options:
  - 59/2 arises from an incorrect scaling (no relation to the 5 from du).
  - 176/5 is an arithmetic error (3×59 = 177, not 176).
  - 88/3 is unrelated to the correct substitution factor and result.

6) Conclusion
The evaluated integral equals 177/5, which matches the provided correct answer.

If 𝑓 is continuous and ∫ 1 243 𝑓 ( 𝑡 ) 𝑑 𝑡 = 59 , evaluate the integral ∫ 1 3 3 𝑡 4 𝑓 ( 𝑡 5 ) 𝑑 𝑡 .单项选择题

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