Derivative and integral of inverse secant | Summary and Q&A

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April 5, 2018
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blackpenredpen
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Derivative and integral of inverse secant

TL;DR

The video demonstrates how to differentiate and integrate the inverse secant function step by step.

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Questions & Answers

Q: How do you differentiate the inverse secant function?

To differentiate inverse secant x, you can start by setting y = inverse secant x, then differentiate both sides with respect to x using the chain rule and implicit differentiation. You will end up with the derivative dy/dx = 1 / (secant(y) * tangent(y)).

Q: How do you integrate the inverse secant function?

The integration of the inverse secant function involves integration by parts and trigonometric substitution. You can set x = secant(theta) and use the identity secant^2(theta) - 1 = tangent^2(theta) to simplify the integral. The final result is x * inverse secant of x - ln| x + sqrt(x^2 - 1)|.

Q: Why is it important to consider the absolute value in the integration solution?

The absolute value is important in the integration solution because the inverse secant function is not defined for negative values. By including the absolute value, we ensure that the result remains valid for both positive and negative values of x.

Q: What is the significance of using trigonometric substitution in the integration process?

Trigonometric substitution is employed in the integration process to simplify the integral and transform it into a form that can be readily integrated. By choosing x = secant(theta), the integral can be simplified using trigonometric identities and techniques.

Summary & Key Takeaways

  • The video explains how to differentiate the inverse secant function with respect to X using the chain rule and implicit differentiation.

  • It also demonstrates how to integrate the inverse secant function using integration by parts and trigonometric substitution.

  • The final result for the differentiation is dy/dx = 1 / (secant(y) * tangent(y)), and for integration is x * inverse secant of x - ln| x + sqrt(x^2 - 1)|.

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