Understanding Integration of Tangent Function

The video tutorial explains how to compute the integral of 3 tangent x dx. It begins by factoring out the constant and discusses two equivalent antiderivatives for the tangent function. The tutorial then demonstrates the substitution method to solve the integral, using the denominator cosine x. It further explains the power property of logarithms to show the equivalence of the two antiderivatives. The video concludes by summarizing the solution and confirming the equivalence of the antiderivatives.

What is the first step in solving the integral of three tangent x dx?

• Use integration by parts

• Apply the chain rule

• Differentiate the function

• Factor out the constant three

Which of the following is an equivalent antiderivative of tangent u du?

• Natural log of absolute value of tangent u

• Negative natural log of absolute value of cosine u

• Negative natural log of absolute value of secant u

• Natural log of absolute value of sine u

What substitution is used to integrate tangent x as sine x over cosine x?

• u = tangent x

• u = secant x

• u = cosine x

• u = sine x

After substitution, what does the integral of one divided by u become?

• Negative u plus c

• Exponential of u plus c

• u squared over 2 plus c

• Natural log of absolute value of u plus c

What is the purpose of factoring out the negative in the substitution method?

• To simplify the integration process

• To convert the integral into a definite integral

• To change the limits of integration

• To eliminate the constant of integration

How is the power property of logarithms used in this integration?

• To simplify the logarithm to a constant

• To change the base of the logarithm

• To raise the logarithm to a power

• To convert a product into a sum

What is the result of raising cosine x to the power of negative one?

• Cotangent x

• Secant x

• Tangent x

• Sine x

What is the final equivalent antiderivative expression for three tangent x dx?

• Three times natural log of absolute value of cosine x plus c

• Three times natural log of absolute value of tangent x plus c

• Three times natural log of absolute value of secant x plus c

• Three times natural log of absolute value of sine x plus c

Why are there two equivalent antiderivatives for the tangent function?

• Because of different integration techniques

• Due to the properties of logarithms

• Because tangent is a periodic function

• Due to the properties of trigonometric identities

What does the integration of tangent x ultimately demonstrate?

• The complexity of trigonometric functions

• The versatility of substitution methods

• The importance of constants in integration

• The relationship between trigonometric and logarithmic functions