U-Substitution (Indefinite Integrals)

After substituting u in ∫ 6x(x² + 1)² dx, what is the new integral?

• ∫ 6u² du

• ∫ 6u³ du

• ∫ 6u du

• ∫ 6u² du + C

What is the purpose of U-Substitution?

• To simplify complex integrals into more manageable forms.

• To find the derivative of a function.

• To solve differential equations directly.

• To evaluate limits of functions.

What is U-Substitution in integration?

• A method to differentiate functions by substituting variables.

• A technique to simplify integration by replacing part of the integrand with a new variable, usually 'u'.

• A process to find limits of integration in definite integrals.

• A strategy to convert integrals into sums for easier calculation.

When should you use U-Substitution?

• When the integrand is a polynomial function.

• When the integrand is a composite function, especially when the derivative of the inner function is present in the integrand.

• When the limits of integration are infinite.

• When the integrand is a trigonometric function.

What is the final answer for ∫ 6x(x² + 1)² dx after back-substituting?

• (x² + 1)³ + C

• (x² + 1)² + C

• 2(x² + 1)³ + C

• 3(x² + 1)² + C

Example of U-Substitution: ∫ 6x(x² + 1)² dx. What is u?

• u = x² + 1

• u = 6x

• u = x²

• u = (x² + 1)²

What is the final answer for ∫ 2x cos(x²) dx after back-substituting?

• sin(x²) + C

• cos(x²) + C

• 2sin(x²) + C

• sin(2x) + C

What is the first step in U-Substitution?

• Identify a part of the integrand to substitute with 'u', typically the inner function of a composite function.

• Choose the limits of integration for the substitution.

• Differentiate the integrand before substitution.

• Integrate the function without substitution.

What is the integral of cos(u) du?

• sin(u) + C

• cos(u) + C

• -sin(u) + C

• tan(u) + C

What is the integral of 6u² du?

• 2u³ + C

• 3u³ + C

• 6u³ + C

• u³ + C