Integral of tanx

TL;DR

In this lesson, we learn the technique of u substitution to integrate the tangent and cotangent functions using the natural log function.

Transcript

in this lesson we're going to talk about the integral of tangent x dx so what do you think we need to do in order to integrate this function the first thing we need to do is convert tangent into sine and cosine so you need to know that tangent is sine divided by cosine so now how can we integrate this expression we need to use a technique known as ... Read More

Key Insights

• ☺️ Tangent x can be converted into sine and cosine form by using the identity tangent x = sine x / cosine x.
• 😄 U substitution is a powerful technique in calculus that simplifies integrals.
• ☺️ The integral of tangent x is equal to the natural log of the secant x plus a constant.
• ☺️ The integral of cotangent x is equal to the natural log of the sine x plus a constant.
• 🧑‍💻 By using the power rule of logs, the expression with a negative sign in front can be modified into the natural log of the reciprocal function.
• 🧑 U substitution involves choosing a u variable, finding its derivative, and solving for dx in terms of du.
• 🧑‍🏭 Canceling out common factors can simplify the integrals of trigonometric functions.

Questions & Answers

Q: How do we integrate the tangent function?

To integrate the tangent function, we first convert it into the sine and cosine form. Then, we use u substitution, setting u equal to the cosine x. By solving for dx, we can express it in terms of du. Finally, we cancel out the sine x and integrate 1/u to obtain the natural log of the secant x plus a constant.

Q: What is the integral of the cotangent function?

Similar to the tangent function, we convert the cotangent into the sine and cosine form. Using u substitution, with u equal to sine x, we find that the integral of the cotangent x is equal to the natural log of the sine x plus a constant.

Q: How do we modify the expression when there is a negative sign in front?

When there is a negative sign in front, we can apply the power rule of logs. By moving the negative sign to the expression inside the log, we can rewrite it as the natural log of the reciprocal function. For example, in the case of the tangent function, we move the negative sign to cosine, resulting in the natural log of the secant x.

Q: Can you summarize the process of u substitution?

U substitution is a technique used to simplify integrals. We choose a suitable u variable that makes the integral more manageable. We find the derivative of u, solve for dx in terms of du, and substitute these values into the integral. Then, we integrate with respect to u and replace u with its original expression to obtain the final result.

Summary & Key Takeaways

• To integrate the tangent function, we convert it into the sine and cosine form and use u substitution.

• The integral of tangent x is equal to the natural log of the secant x plus a constant.

• To integrate the cotangent function, we again use u substitution and find that the integral is equal to the natural log of the sine x plus a constant.