What Are the Indefinite Integrals of Hyperbolic Functions?

TL;DR

The indefinite integrals of hyperbolic functions are defined as follows: the integral of hyperbolic cosine is hyperbolic sine, the integral of hyperbolic secant is hyperbolic tangent, and the integral of hyperbolic cosecant squared is negative cotangent. Using techniques like U substitution can simplify these integrations, and it's important to handle negative signs correctly for hyperbolic secant and hyperbolic cotangent.

Transcript

in this lesson we're going to cover the integrals of hyperbolic functions so if you have a sheet of paper you want to take some notes let's go ahead and begin the indefinite integral of hyperbolic cosine is hyperbolic sine it's the opposite of the derivatives the derivative of hyperbolic sine is cosine the antiderivative of hyperbolic sine is hyper... Read More

Key Insights

• ❓ The indefinite integrals of hyperbolic functions have specific formulas.
• 👨‍💼 The antiderivative of hyperbolic sine is hyperbolic cosine.
• 👨‍💼 The derivative of hyperbolic sine is cosine.
• 😇 There are different formulas for integrating hyperbolic cosecant squared, secant tan, and hyperbolic cosecant cotangent.
• 😄 U substitution can be used to simplify the integration process.
• 🤘 Pay attention to any negative signs when integrating hyperbolic secant and hyperbolic cotangent.
• ✊ The power rule can be applied when integrating certain hyperbolic functions.

Questions & Answers

Q: What is the indefinite integral of hyperbolic cosine 2x?

The indefinite integral of hyperbolic cosine 2x is one-half hyperbolic sine of 2x plus a constant.

Q: What is the indefinite integral of hyperbolic sine 5x plus 4?

The indefinite integral of hyperbolic sine 5x plus 4 is 5 hyperbolic cosine 5x plus a constant.

Q: What is the indefinite integral of hyperbolic cotangent x?

The indefinite integral of hyperbolic cotangent x is the natural log of hyperbolic sine x plus a constant.

Q: How do you integrate the cube of hyperbolic secant times hyperbolic tangent?

To integrate the cube of hyperbolic secant times hyperbolic tangent, use the substitution method and let u be equal to secant x. The resulting integral is -1/3 hyperbolic secant cubed x plus a constant.

Summary & Key Takeaways

• The indefinite integral of hyperbolic cosine is hyperbolic sine, the opposite of its derivative.

• The indefinite integral of hyperbolic secant is hyperbolic tangent.

• The indefinite integral of hyperbolic cosecant squared is negative cotangent.

• The indefinite integral of secant tan is negative secant.

• The indefinite integral of hyperbolic cosecant cotangent is negative cosecant.