Antiderivatives and Integration Techniques

The video tutorial explains how to compute the integral of 2 tangent 9x dx using substitution. It introduces two integral formulas for tangent u and demonstrates the substitution method by setting u = 9x. The process involves simplifying the integral, performing integration, and finding the antiderivative. The tutorial also discusses an alternative formula for expressing the antiderivative.

What is the integral of tangent u du according to the first formula?

• Natural log of the absolute value of cosine u plus c

• Negative natural log of the absolute value of sine u plus c

• Natural log of the absolute value of sine u plus c

• Negative natural log of the absolute value of cosine u plus c

In the substitution method, what is u equal to?

• x

• 2x

• x/9

• 9x

After substitution, what does dx equal?

• 2 du

• du

• 9 du

• 1/9 du

What constant is factored out when rewriting the integral with respect to u?

• 2/9

• 9/2

• 1/2

• 1/9

What is the antiderivative of the integral in terms of x using the first formula?

• 2/9 times natural log of the absolute value of cosine 9x plus c

• Negative 2/9 times natural log of the absolute value of cosine 9x plus c

• Negative 2/9 times natural log of the absolute value of sine 9x plus c

• 2/9 times natural log of the absolute value of sine 9x plus c

Using the second integration formula, how can the antiderivative be expressed?

• Negative 2/9 times natural log of the absolute value of cosine 9x plus c

• Positive 2/9 times natural log of the absolute value of cosine 9x plus c

• Negative 2/9 times natural log of the absolute value of secant 9x plus c

• Positive 2/9 times natural log of the absolute value of secant 9x plus c

What are the two equivalent antiderivatives for the integral of tangent u du?

• Natural log of cosine u and negative natural log of secant u

• Negative natural log of sine u and natural log of cosine u

• Natural log of sine u and negative natural log of cosine u

• Negative natural log of cosine u and natural log of secant u