Why Can't the Integral from 1 to x of $\sqrt{1+{t}^{4}}$ be Replaced with x?

  • Context: MHB 
  • Thread starter Thread starter karush
  • Start date Start date
  • Tags Tags
    Integral
Click For Summary
SUMMARY

The integral from 1 to x of the function $\sqrt{1+t^4}$ cannot be simply replaced with x due to the nature of definite integrals. The function defined as $F(x) = \int_{1}^{x} \sqrt{1+t^4} \, dt$ requires evaluation at the upper limit x, which results in the derivative $F'(x) = \sqrt{1+x^4}$. This highlights the importance of correctly applying the Fundamental Theorem of Calculus when differentiating integrals.

PREREQUISITES
  • Understanding of definite integrals
  • Familiarity with the Fundamental Theorem of Calculus
  • Knowledge of derivatives and their properties
  • Basic proficiency in calculus notation
NEXT STEPS
  • Study the Fundamental Theorem of Calculus in detail
  • Practice finding derivatives of definite integrals
  • Explore advanced integration techniques for complex functions
  • Learn about the implications of variable substitution in integrals
USEFUL FOR

Students and educators in calculus, mathematicians focusing on integral calculus, and anyone interested in understanding the nuances of differentiating integrals.

karush
Gold Member
MHB
Messages
3,240
Reaction score
5
got really hefty answers with the calculator but can t be replaced with x

$$F\left(x\right)=\int_{1}^{x}\sqrt{1+{t}^{4}} \,dt$$

I assume it is just the derivative

$\sqrt{1+{t}^{4}}$
 
Last edited:
Physics news on Phys.org
karush said:
got really hefty answers with the calculator but can t be replaced with x

$$F\left(x\right)=\int_{1}^{x}\sqrt{1+{t}^{4}} \,dt$$

I assume it is just the derivative

$\sqrt{1+{t}^{4}}$
You haven't said what the problem is. If it is to find the derivative $F'(x)$ then the answer is $\sqrt{1+{x}^{4}}$ (with an $x$, not a $t$).
 

Similar threads

  • · Replies 6 ·
Replies
6
Views
2K
  • · Replies 4 ·
Replies
4
Views
4K
  • · Replies 12 ·
Replies
12
Views
3K
  • · Replies 20 ·
Replies
20
Views
4K
  • · Replies 3 ·
Replies
3
Views
3K
  • · Replies 2 ·
Replies
2
Views
3K
  • · Replies 3 ·
Replies
3
Views
3K
  • · Replies 19 ·
Replies
19
Views
4K
  • · Replies 5 ·
Replies
5
Views
2K
  • · Replies 2 ·
Replies
2
Views
1K