What is the derivative of $$u = \tan(2x)$$ ?

$$\frac{du}{dx} = 2\text{sec}^2(2x)$$ .

What is the integral of $$\frac{\text{sin}(2x)}{\text{cos}^4(2x)}dx$$ ?

$$-\frac{1}{3\text{cos}^3(2x)} + C$$ .

What is the integral of $$\frac{12x}{(3x^2+1)^3}dx$$ ?

$$-\frac{1}{(3x^2+1)^2} + C$$ .

What is the relationship between u-substitution and the chain rule?

U-substitution is essentially a backward application of the chain rule, allowing us to integrate functions that are compositions of other functions.

What is the first step in u-substitution?

Identify a part of the integrand to substitute with u, typically a function whose derivative is also present in the integrand.

What is the importance of the constant C in integration?

The constant C represents the family of antiderivatives, as indefinite integrals can differ by a constant.

What is the integral of $$\frac{1}{(3x^2+1)^2}dx$$ ?

$$\frac{1}{3}(3x^2+1)^{-1} + C$$ .

U Substitution With Manipulation

u-substitution with manipulation

Flashcard

•

Mathematics

•

12th Grade

•

Practice Problem

•

Hard

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FLASHCARD QUESTION

Front

What is u-substitution in integration?

Back

U-substitution is a method used to simplify integrals by substituting a part of the integrand with a new variable, u, making the integral easier to solve.

FLASHCARD QUESTION

Front

What is the purpose of choosing u in u-substitution?

Back

Choosing u simplifies the integral, often transforming it into a basic form that can be easily integrated.

FLASHCARD QUESTION

Front

Back

FLASHCARD QUESTION

Front

Back

FLASHCARD QUESTION

Front

What is the chain rule in calculus?

Back

The chain rule is a formula for computing the derivative of the composition of two or more functions.

FLASHCARD QUESTION

Front

How do you apply the chain rule in u-substitution?

Back

When using u-substitution, the chain rule helps to relate the derivative of u back to the original variable, allowing for proper integration.

FLASHCARD QUESTION

Front

Back

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