U substitution for differentiation?

Thread starter kurious
Start date Aug 5, 2004
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Differentiation Substitution U substitution

In summary, U substitution for differentiation is a method used in calculus to solve integrals by substituting a variable u for an expression within the integral. To perform U substitution, steps include identifying the expression, finding its derivative, substituting u and du into the integral, solving in terms of u, and substituting back in the original variable for u. This method is useful for solving complex integrals and identifying patterns. However, common mistakes include choosing an incorrect expression, forgetting to take the derivative, not simplifying before substituting, and not checking the final answer. It cannot be used for integrals involving trigonometric functions or logarithms.

Aug 5, 2004

kurious

How do I evaluate:

d/dt sqrt [ t^4 + t^2 ]= 0
to get a max/min value.

can I make a u substitution of some sort?

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Aug 5, 2004

HallsofIvy

Science Advisor

Homework Helper

Yes, that's basically just the "chain rule".

Let u= t⁴+ t²

then d sqrt(t⁴+ t²)/dt= d u^1/2/du* du/dt
= (1/2)u^-1/2 * (4t³+ 2t)
= (1/2)(t⁴+t²)^-1/2)*(4t³+ 2t)

Aug 5, 2004

kurious

Thanks for your help.

What is U substitution for differentiation?

U substitution for differentiation is a method used in calculus to solve integrals. It involves substituting a variable u for an expression within the integral to simplify the problem.

How do you perform U substitution for differentiation?

To perform U substitution for differentiation, follow these steps:

Identify an expression within the integral that can be substituted with u.
Let u equal that expression and find its derivative, du.
Substitute u and du into the integral.
Solve the integral in terms of u.
Finally, substitute back in the original variable for u.

Why is U substitution for differentiation useful?

U substitution for differentiation is useful because it allows us to solve more complex integrals by reducing them to simpler forms. It also helps us to identify patterns and make connections between different types of integrals.

What are some common mistakes when using U substitution for differentiation?

Some common mistakes when using U substitution for differentiation include:

Choosing an incorrect expression to substitute for u.
Forgetting to take the derivative of u.
Forgetting to substitute back in the original variable for u.
Not simplifying the integral before substituting u.
Not checking the final answer by differentiating it.

Can U substitution for differentiation be used for all integrals?

No, U substitution for differentiation can only be used for integrals in which there is a nested function that can be simplified by substituting with u. It cannot be used for integrals involving trigonometric functions or logarithms.