Antiderivatives and Derivatives Concepts

This lesson covers evaluating indefinite integrals using u-substitution. It provides two examples: one with a polynomial integrand and another involving secant squared. The process involves identifying parts of the integrand to substitute, determining the differential, and applying integration formulas. The video emphasizes recognizing patterns and simplifying integrals to find antiderivatives.

What is the primary method discussed in this lesson for evaluating indefinite integrals?

• Partial fraction decomposition

• Trigonometric substitution

• U-substitution

• Integration by parts

In the first example, what is chosen as the substitution variable u?

• 3x

• Negative 4x squared

• e to the power of 3x

• Secant squared

What is the derivative of negative 4x squared with respect to x?

• 8x

• Negative 8x

• 4x

• Negative 2x

What is the antiderivative of e to the power of u with respect to u?

• u squared over 2

• e to the power of u plus c

• e to the power of u minus c

• u times e to the power of u

In the second example, what function is involved in the integrand?

• Secant squared

• Cosine

• Sine

• Tangent

What substitution is made for u in the second example?

• 3x

• Negative 4x squared

• e to the power of 3x

• Secant squared

What is the derivative of e to the power of 3x with respect to x?

• 3x

• e to the power of 3x

• e to the power of x

• 3e to the power of 3x

What is the antiderivative of secant squared u with respect to u?

• Cosecant u plus c

• Secant u plus c

• Cotangent u plus c

• Tangent u plus c

What is the final form of the antiderivative in the second example?

• Two-thirds times tangent of e to the power of 3x plus c

• Three times tangent of e to the power of 3x plus c

• One-third times tangent of e to the power of 3x plus c

• Two times secant of e to the power of 3x plus c

How is the family of antiderivatives often identified?

• Small f of x

• Big F of x

• Derivative of x

• Integral of x