What is the integral of $$6x(3x^2 + 4)^4dx$$ using u-substitution?

$$\frac{(3x^2 + 4)^5}{5} + C$$ .

Solve the integral $$\text{If } u = x^3 - 2, \text{ then } \text{integrate } 3x^2\text{d}x$$ .

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

What is the integral of $$6x(x^2 + 1)^2dx$$ using u-substitution?

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

Explain the importance of the constant of integration (C).

The constant of integration (C) represents the family of antiderivatives and accounts for the fact that indefinite integrals can differ by a constant.

How do you identify the substitution variable in u-substitution?

Look for a function within the integrand whose derivative also appears in the integrand.

What is the purpose of u-substitution in calculus?

U-substitution simplifies complex integrals, making them easier to evaluate.

Provide an example of a function suitable for u-substitution.

$$\text{Example: } \text{Integrate } \text{d}x \text{ where } u = x^2 + 1$$ .

What is the general process for performing u-substitution?

1. Choose a substitution $$u = g(x)$$ . 2. Compute $$du = g$$ . 3. Rewrite the integral in terms of u. 4. Integrate. 5. Substitute back to x.

12/10/24 u-Substitution Flashcards

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1.

FLASHCARD QUESTION

Front

Define u-substitution in integration.

Back

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

2.

FLASHCARD QUESTION

Front

What is the formula for u-substitution?

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3.

FLASHCARD QUESTION

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4.

FLASHCARD QUESTION

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5.

FLASHCARD QUESTION

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6.

FLASHCARD QUESTION

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7.

FLASHCARD QUESTION

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