Integrating Tangent and Secant Functions

The video tutorial explains how to integrate tangent cubed by splitting it into two parts: tangent squared and tangent. It uses trigonometric identities and u-substitution to solve the integrals separately. The first integral involves replacing tangent with a u-variable, while the second integral converts tangent into sine over cosine. The final answer is presented in two equivalent forms, highlighting the use of natural logarithms and trigonometric identities.

What is the initial strategy to integrate tangent cubed?

• Use partial fraction decomposition

• Split into tangent squared and tangent

• Directly integrate using a table

• Apply integration by parts

Which identity is used to express tangent squared in terms of secant squared?

• 1 + tan^2(x) = sec^2(x)

• tan^2(x) = 1 - sec^2(x)

• tan^2(x) = sec^2(x) - 1

• tan^2(x) = sec^2(x) + 1

What substitution is used to integrate secant squared times tangent?

• u = cos(x)

• u = tan(x)

• u = sec(x)

• u = sin(x)

What is the integral of u with respect to u?

• u^2/2 + C

• ln|u| + C

• 1/u + C

• u + C

How is tangent expressed to facilitate its integration?

• As cosine over sine

• As sine over cosine

• As secant over cosine

• As secant over sine

What substitution is used to integrate tangent?

• u = sec(x)

• u = cos(x)

• u = tan(x)

• u = sin(x)

What is the antiderivative of 1/u?

• u + C

• 1/u + C

• ln|u| + C

• u^2/2 + C

What is the final form of the integral of tangent cubed?

• 1/2 tan^2(x) + ln|cos(x)| + C

• ln|tan(x)| + C

• 1/2 tan^2(x) - ln|sec(x)| + C

• tan^3(x)/3 + C

What is an alternative form of the final answer?

• 1/2 tan^2(x) + ln|sec(x)| + C

• 1/2 tan^2(x) - ln|cos(x)| + C

• 1/2 tan^2(x) + ln|cos(x)| + C

• tan^3(x)/3 - ln|sec(x)| + C

What is the relationship between secant and cosine used in the final answer?

• sec(x) = sin(x)/cos(x)

• sec(x) = cos(x)

• sec(x) = 1/cos(x)

• sec(x) = 1/sin(x)