Evaluate. (Be sure to check by differentiating)
[tex]$\int\left(6 x^4+8\right) t^3 d t$[/tex]
A. [tex]$u=t^3+8$[/tex]
B. [tex]$u=4^3$[/tex]
C. [tex]$u=6 t+8$[/tex]
D. [tex]$u=6 t^4+8$[/tex]
Write the integral in terms of u.
[tex]$\int\left(6 t^4+8\right) t^3 d t=\int(\square) d u$[/tex]
(Type an exact answer. Use parentheses to clearly denote the argument of each function.)
Answer (2)
Analyze the given integral and substitution options.
Evaluate each substitution to determine which simplifies the integral.
Choose the substitution u = 6 t 4 + 8 , which gives d u = 24 t 3 d t .
Rewrite the integral in terms of u : ∫ 24 u d u .
Explanation
Analyzing the Problem and Substitution Options We are given the integral ∫ ( 6 t 4 + 8 ) t 3 d t and asked to rewrite it in terms of u using a suitable substitution from the options provided. The options are: A. u = t 3 + 8 B. u = t 4 C. u = 6 t + 8 D. u = 6 t 4 + 8 We need to determine which substitution simplifies the integral most effectively.
Evaluating Each Substitution Let's analyze each substitution:
Option A: u = t 3 + 8 . Then d u = 3 t 2 d t . This doesn't directly simplify the integral since we have t 3 d t in the integral, not t 2 d t .
Option B: u = t 4 . Then d u = 4 t 3 d t , so t 3 d t = 4 1 d u . Substituting this into the integral gives ∫ ( 6 t 4 + 8 ) t 3 d t = ∫ ( 6 u + 8 ) 4 1 d u = 4 1 ∫ ( 6 u + 8 ) d u . This looks promising.
Option C: u = 6 t + 8 . Then d u = 6 d t . This doesn't directly simplify the integral since we have t 3 d t in the integral, not just d t .
Option D: u = 6 t 4 + 8 . Then d u = 24 t 3 d t , so t 3 d t = 24 1 d u . Substituting this into the integral gives ∫ ( 6 t 4 + 8 ) t 3 d t = ∫ u 24 1 d u = 24 1 ∫ u d u . This also looks promising.
Rewriting the Integral with the Chosen Substitution Comparing options B and D, option D, u = 6 t 4 + 8 , leads to a simpler integral in terms of u . Therefore, we choose substitution D: u = 6 t 4 + 8 and d u = 24 t 3 d t , so t 3 d t = 24 1 d u .
Now, we rewrite the integral: ∫ ( 6 t 4 + 8 ) t 3 d t = ∫ u 24 1 d u = 24 1 ∫ u d u The integral in terms of u is ∫ ( 6 t 4 + 8 ) t 3 d t = ∫ 24 u d u .
Final Answer Thus, the integral in terms of u is: ∫ 24 u d u Therefore, the expression that goes in the blank is 24 u .
Examples
Imagine you're designing a self-watering plant pot. To optimize the water release, you need to calculate the total water released over time, which can be modeled by an integral. By using substitution, you can simplify the integral and easily find the total water released, helping you design an efficient and sustainable plant pot.
The integral ∫ ( 6 t 4 + 8 ) t 3 d t can be rewritten in terms of u using the substitution u = 6 t 4 + 8 , leading to the expression ∫ 24 u d u . The substitution simplifies the evaluation process significantly. Thus, the correct choice is option D.
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