my favorite calculus 2 trigonometry integral (two results off by a constant)

TL;DR

Two substitutions are shown for solving the integral of tan(x)*sec^2(x), resulting in different but correct solutions.

Transcript

so good okay i'm going to show you guys two substitutions  to integral of tan(x)*sec^2(x) let's do the   obvious one first for this one right here  let me just say that u equal to tangent x   how come because i know right away when i  differentiate this i get the derivative of tangent   x which is secant squared x dx right and you see i  can take t... Read More

Key Insights

• 💦 Letting u equal to either tan(x) or sec(x) provides substitutions to simplify the integral of tan(x)*sec^2(x).
• 🥺 Both substitutions lead to correct solutions, but they may differ in their constant values.
• ❓ The solutions obtained through different substitutions can be adjusted by assigning different constants (c1 and c2) to account for this discrepancy.
• ❓ The concept of "off by a constant" is illustrated through the derivation of the secant and tangent identity.

Questions & Answers

Q: How does the first substitution for the integral of tan(x)*sec^2(x) work?

By letting u=tan(x), we can convert the integral into the u world, making it easier to solve. The resulting solution is 1/2*tan^2(x).

Q: What is the second substitution for the integral of tan(x)*sec^2(x)?

The second substitution involves letting u=sec(x), which simplifies the integral further. The solution in this case is 1/2*sec^2(x).

Q: Why do the two solutions for the integral look different?

The difference in constant values (c1 and c2) leads to the different-looking solutions. However, both solutions are correct, as they are only off by a constant.

Q: What is meant by the term "off by a constant"?

The result of an integration can differ by a constant value. The constant can be represented as c1 and c2 in the two solutions, reflecting the variation in integrating constant.

Summary & Key Takeaways

• The video demonstrates two substitutions for solving the integral of tan(x)*sec^2(x).

• The first substitution involves letting u=tan(x) and simplifying the integral to become 1/2*tan^2(x).

• The second substitution involves letting u=sec(x) and simplifying the integral to become 1/2*sec^2(x).