U-Substitution and Integration Techniques

This video tutorial explains how to find the indefinite integral of x divided by the square root of x plus 1 using u-substitution. The process involves setting up the substitution, transforming the integral, separating it into two parts, and applying the power rule to integrate. The tutorial concludes with simplifying the result and deriving the final answer.

What is the initial expression we are trying to integrate in this video?

• x plus the square root of x plus 1

• x divided by the square root of x plus 1

• x squared plus 1

• x times the square root of x plus 1

What substitution is made for u in the u-substitution process?

• u = square root of x

• u = x minus 1

• u = x plus 1

• u = x

After substituting, what expression do we get for x in terms of u?

• x = u plus 1

• x = u minus 1

• x = u divided by 1

• x = u times 1

How is the integral separated into two parts?

• By dividing u and negative 1 by the square root of u

• By adding the integrals

• By multiplying the integrals

• By subtracting the integrals

What is the exponent of u after dividing u by the square root of u?

• 3/2

• 1/2

• -1/2

• 1

What rule is used to integrate the expressions after simplification?

• Chain Rule

• Power Rule

• Quotient Rule

• Product Rule

What is the result of adding 1/2 and 1 in the integration process?

• 2

• 3/2

• 1

• 5/2

What is the final expression for the integral after substituting back to x?

• 1/3 times (x plus 1) raised to the 3/2 minus 2 times (x plus 1) raised to the 1/2 plus c

• 2 times (x plus 1) raised to the 3/2 minus 3 times (x plus 1) raised to the 1/2 plus c

• 2/3 times (x plus 1) raised to the 3/2 minus 2 times (x plus 1) raised to the 1/2 plus c

• 3/2 times (x plus 1) raised to the 2/3 minus 1 times (x plus 1) raised to the 1/2 plus c

What constant is added to the final integrated expression?

• c

• f

• d

• e

What is the purpose of multiplying the top and bottom by two-thirds in the simplification process?

• To eliminate fractions

• To change the variable

• To find the derivative

• To simplify the expression