# Integral of Hyperbolic Functions | Summary and Q&A

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April 19, 2023
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The Organic Chemistry Tutor
Integral of Hyperbolic Functions

## TL;DR

This lesson covers the indefinite integrals of hyperbolic functions, such as hyperbolic cosine, hyperbolic sine, hyperbolic secant, hyperbolic cosecant, and hyperbolic cotangent.

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### 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.