integral of e^x*sinx*cosx, (with Jordan 14s Last Shot)

TL;DR

Simplify the integral e^x * e^x * cos(x) using trigonometric identities.

Transcript

this integral should be crazy enough edits by the look we have the integral e to the x times e x times cos x we have done a similar integral in the past but it was just the integral of e 2x times x there was no cos x and to do this integral here we could just use the integration by parts and right here if you want to use integration by parts as how... Read More

Key Insights

• ❓ By using trigonometric identities, the integral e^x * e^x * cos(x) can be simplified.
• ❓ The trigonometric identity sin(2x) = 2sin(x)cos(x) is used to simplify the integral.
• ❓ After simplification, the integral becomes e^x * 1/2 * sin(2x).
• 🥳 Integration by parts is then used to further simplify the integral.

Questions & Answers

Q: How can the integral e^x * e^x * cos(x) be simplified?

The integral can be simplified by using the trigonometric identity sin(2x) = 2sin(x)cos(x) to replace sin(x) * cos(x) with 1/2 * sin(2x).

Q: What is the result of replacing sin(x) * cos(x) with 1/2 * sin(2x)?

By replacing the term, the integral becomes e^x * 1/2 * sin(2x), which is easier to integrate.

Q: How can the integral e^x * 1/2 * sin(2x) be further simplified?

The integral can be further simplified using integration by parts, by integrating sin(2x) and differentiating e^x.

Q: What is the final answer to the integral e^x * e^x * cos(x)?

After simplifying and integrating, the final answer to the integral is -1/5 * e^x * cos(2x) + 1/10 * e^x * sin(2x) + C.

Summary & Key Takeaways

• The integral e^x * e^x * cos(x) can be simplified by using the identity sin(2x) = 2sin(x)cos(x).

• By replacing sin(x) * cos(x) with 1/2 * sin(2x), the integral becomes e^x * 1/2 * sin(2x).

• The integral e^x * 1/2 * sin(2x) is then further simplified using integration by parts.