Antiderivatives and Rational Exponents

The video tutorial explains how to determine antiderivatives using the power rule of integration. It provides two examples: finding the antiderivative of x^(2/3) and the square root of x. The process involves adding 1 to the exponent, dividing by the new exponent, and adding a constant. The video also verifies the results by differentiating the obtained antiderivatives to ensure they match the original functions.

What is the first step in finding the antiderivative of x to the power of 2/3?

• Subtract 1 from the exponent

• Add 1 to the exponent

• Divide the exponent by 2

• Multiply the exponent by 2

After adding 1 to the exponent, what is the next step in finding the antiderivative?

• Add a constant to the exponent

• Divide by the new exponent

• Subtract the new exponent

• Multiply by the new exponent

What is the reciprocal of 5/3 used for in the calculation?

• To subtract from the exponent

• To divide the original function

• To multiply with the result

• To add to the constant C

How do you verify the antiderivative of x to the power of 2/3?

• By adding a constant to the result

• By multiplying the result by 2

• By differentiating the result

• By integrating the result

What is the first step in finding the antiderivative of the square root of x?

• Rewrite it as a rational exponent

• Multiply by 2

• Add a constant

• Subtract 1 from the exponent

What is the rational exponent form of the square root of x?

• x^(3/2)

• x^(1/3)

• x^(1/2)

• x^(2/3)

What is the next step after rewriting the square root of x as a rational exponent?

• Divide by the exponent

• Add 1 to the exponent

• Multiply by the exponent

• Subtract 1 from the exponent

How do you verify the antiderivative of the square root of x?

• By differentiating the result

• By integrating the result

• By adding a constant to the result

• By multiplying the result by 3

What is the final form of the antiderivative of the square root of x?

• 5/3 x^(3/2) + C

• 1/2 x^(5/3) + C

• 3/2 x^(2/3) + C

• 2/3 x^(3/2) + C

What is the purpose of adding a constant C in the antiderivative?

• To divide the result

• To adjust the exponent

• To account for the constant of integration

• To multiply the result