How to Integrate csc(x)sec(x) Step by Step

TL;DR

To integrate csc(x)sec(x), rewrite it as sin²(x) + cos²(x) over sin(x)cos(x). This allows you to simplify the expression, leading to the integral of tan(x) and cot(x), resulting in ln|tan(x)| + C. Applying trigonometric identities efficiently simplifies the integration process.

Transcript

okay let's integrate cose x * SEC X how can we do this we are not really sure about how to integrate this right secant x times cosecant X how about let's just try to write everything in terms of s and cosine this is 1 / sin x and this is 1 / cosine X and let's do that first so this is the same as integrating I can put one on the top over this terms... Read More

Key Insights

• 😑 Rewriting the given expression in terms of sine and cosine allows for more straightforward manipulation and integration.
• 😑 The trigonometric identity sin^2(x) + cos^2(x) = 1 is a useful tool in expanding trigonometric expressions.
• ❓ Integrating tangent and cotangent functions can be accomplished using known formulas.

Questions & Answers

Q: How does the speaker begin to integrate the expression cos(x) * sec(x)?

The speaker starts by rewriting cos(x) * sec(x) as 1/sin(x) * 1/cos(x), or 1/(sin(x) * cos(x)).

Q: Why does the speaker rewrite the expression in terms of sine and cosine?

By expressing cos(x) * sec(x) in terms of sin(x) and cos(x), the speaker aims to simplify the expression further and make it more amenable to integration techniques.

Q: What trigonometric identity does the speaker use to expand the expression?

The speaker utilizes the identity sin^2(x) + cos^2(x) = 1 to rewrite 1/(sin(x) * cos(x)) as sin(x)/cos(x) + cos(x)/sin(x).

Q: How does the speaker integrate the expanded expression?

The speaker applies the known integration formulas for tangent and cotangent functions, which are Ln|sec(x)| + C and Ln|sin(x)| + C, respectively.

Summary & Key Takeaways

• The video demonstrates how to rewrite cos(x) * sec(x) in terms of sine and cosine.

• The speaker applies a trigonometric identity to expand the expression into sin^2(x) + cos^2(x).

• By canceling out terms and utilizing known integration formulas, the speaker obtains the final result of Ln|tan(x)| + C.