Integration Techniques and U-Substitution

The video tutorial explains how to evaluate indefinite integrals using integration formulas. It covers two examples: one involving an integral with a square root in the denominator, solved using the arc sine formula, and another with a sum in the denominator, solved using the arctangent formula. The tutorial demonstrates the process of rewriting integrals to fit standard forms, performing u-substitution, and applying integration formulas to find antiderivatives.

What is the form of the integration formula used for the first example?

• Integral of du times (a^2 + u^2)

• Integral of du times (a^2 - u^2)

• Integral of du divided by (a^2 + u^2)

• Integral of du divided by the square root of (a^2 - u^2)

In the first example, what is the value of 'a' after rewriting the integral?

• 4

• 2

• 5

• 3

What substitution is performed for 'u' in the first example?

• u = 3x

• u = 2x

• u = 4x

• u = 5x

What is the final result of the first example after applying the integration formula?

• 2 times the arc tangent of (3x/2) plus C

• 1/2 times the arc sine of (3x/2) plus C

• 2 times the arc sine of (3x/2) plus C

• 1/2 times the arc tangent of (3x/2) plus C

What is the integration formula used for the second example?

• Integral of du times (a^2 - u^2)

• Integral of du divided by the square root of (a^2 - u^2)

• Integral of du divided by (a^2 + u^2)

• Integral of du times (a^2 + u^2)

In the second example, what is the value of 'a' after rewriting the integral?

• 4

• 5

• 2

• 3

What substitution is performed for 'u' in the second example?

• u = 3x

• u = 2x

• u = 5x

• u = 4x

What is the final result of the second example after applying the integration formula?

• 1/10 times the arc sine of (4x/5) plus C

• 1/10 times the arc tangent of (4x/5) plus C

• 1/5 times the arc sine of (4x/5) plus C

• 1/5 times the arc tangent of (4x/5) plus C

What is the derivative of the function family obtained in the second example?

• The reciprocal of the integrand function

• The square of the integrand function

• The given integrand function

• The integral of the integrand function

What is the purpose of performing u-substitution in these examples?

• To simplify the integrand function

• All of the above

• To change the variable of integration

• To match the integrand with a known formula