How to Solve the Indefinite Integral of x cos²(x) tan(x)

TL;DR

To solve the indefinite integral of x cos²(x) tan(x), first simplify the expression by canceling a cosine term and using the double-angle formula to rewrite it as one-half sine 2x. Then apply integration by parts to find the integral, resulting in the final answer of -1/4 x cos(2x) + 1/8 sin(2x) + C.

Transcript

what is the indefinite integral of x cosine squared x times tangent x feel free to try this problem if you want to go ahead and pause the video take a minute to work on it now the first thing we need to do is simplify if possible cosine squared is basically cosine x times cosine x tangent is sine divided by cosine so at this point we can cancel a c... Read More

Key Insights

• 😑 Simplification of the expression is an important step in finding the indefinite integral.
• 👨‍💼 The double-angle formula provides a useful substitution for sine and cosine terms.
• 🥳 Integration by parts is a valuable technique for solving complex integrals.
• ❓ Understanding the properties and derivatives of trigonometric functions is crucial in solving integral problems.

Questions & Answers

Q: How do we simplify the expression x cos^2(x) tan(x)?

We can simplify the expression by canceling out a cosine term, which transforms it into x sin(x) cos(x).

Q: What is the double-angle formula used in this problem?

The double-angle formula used is sine 2x = 2sin(x)cos(x), which is derived from sine squared x + cosine squared x = 1.

Q: What is the formula for integration by parts?

The formula for integration by parts is ∫udv = uv - ∫vdu, where u and v are functions of x.

Q: How is integration by parts applied in this problem?

In this problem, x is chosen as the u variable and sin(2x)dx is chosen as dv. By substitution and integration, the final result is obtained.

Summary & Key Takeaways

• The first step is to simplify the expression by cancelling out a cosine term.

• The double-angle formula is used to replace sine and cosine terms, resulting in the expression one-half sine 2x.

• Integration by parts is then applied to find the integral of the simplified expression.