Indefinite Integrals and U-Substitution

The video tutorial explains how to evaluate indefinite integrals using u substitution. It covers two examples: integrating 6 cosecant 4x times cotangent 4x and 14 secant squared 7x. The process involves identifying the inner function, determining the differential, and rewriting the integral in terms of u. The tutorial concludes with the final expressions in terms of x, highlighting the family of functions whose derivatives match the original integrands.

What is the purpose of evaluating indefinite integrals?

• To calculate limits

• To find the area under a curve

• To determine the antiderivative

• To solve differential equations

In the first example, why is u-substitution necessary?

• To eliminate the constant factor

• To simplify the integration process

• Because the integral is too complex

• Because the input is 4x instead of x

What is the value of differential u in the first example?

• 4 times dx

• 6 times dx

• dx

• 1/4 times dx

How is dx replaced in the first example after u-substitution?

• With 1/6 du

• With 6 du

• With 4 du

• With 1/4 du

What is the final result of the first example in terms of x?

• Three halves secant 4x plus c

• Negative three halves secant 4x plus c

• Three halves cosecant 4x plus c

• Negative three halves cosecant 4x plus c

What is the basic integration formula for secant squared x dx?

• Cosecant x plus c

• Cotangent x plus c

• Tangent x plus c

• Secant x plus c

In the second example, what is u equal to?

• 14x

• 7x

• 4x

• x

How is dx replaced in the second example after u-substitution?

• With 1/14 du

• With 1/7 du

• With 7 du

• With 14 du

What is the final result of the second example in terms of x?

• Two times cotangent of 7x plus c

• Two times cosecant of 7x plus c

• Two times secant of 7x plus c

• Two times tangent of 7x plus c

What is the family of functions whose derivative is equal to the integrand function of 14 secant squared 7x?

• Two times secant of 7x plus c

• Two times cosecant of 7x plus c

• Two times cotangent of 7x plus c

• Two times tangent of 7x plus c