Integration into Inverse trigonometric functions using Substitution | Summary and Q&A

TL;DR

Learn how to integrate functions using the formulas for inverse trig functions, such as arc sine, arc tangent, and arc secant.

Key Insights

• 🫠 Three formulas for integrating functions involving inverse trig functions are introduced: arc sine, arc tangent, and arc secant.
• 🔨 Completing the square is a useful tool for simplifying functions and identifying the appropriate formula for integration.
• 🥘 Substituting appropriate values of u and a into the formulas allows for the integration of a given function.
• ❓ The constant of integration is crucial in the process of integration.
• ❓ Each formula has its own specific conditions and restrictions, resulting in different approaches to solving integration problems.
• 🆘 The substitutions made in these integration problems help simplify the functions and make them compatible with the inverse trigonometric formulas.
• ⚾ The process of integration involves identifying the correct formula based on the given function and then substituting appropriate values to obtain the antiderivative.

Transcript

in this video we're going to talk about how to integrate functions that lead to inverse trig functions now there's three formulas you need to know the first one the antiderivative of d u divided by the square root of a squared minus u squared this is equal to arc sine u over a plus c so that's the first equation you want to know the next one looks ... Read More

Questions & Answers

Q: What are the three formulas for integrating functions involving inverse trig functions?

The first formula is the antiderivative of (du / sqrt(a^2 - u^2)), which gives the arc sine of u over a plus a constant. The second formula is the antiderivative of (du / (a^2 + u^2)), which gives 1 over a times the arc tangent of u over a plus a constant. The third formula is the antiderivative of (du / (u * sqrt(u^2 - a^2))), which gives 1 over a times the arc secant of u over a plus a constant.

Q: How do you use the formulas to integrate a given function?

To integrate a function, you need to identify the appropriate formula based on the form of the function. Then, substitute the values of u and a into the formula and simplify to obtain the antiderivative. Finally, add a constant of integration.

Q: What is the antiderivative of dx / sqrt(16 - x^2)?

First, identify u^2 and a^2, which are x^2 and 16, respectively. Substitute u and a into the formula for arc sine, giving arc sine(x/4) + C as the antiderivative.

Q: How do you integrate 3 / (25 + x^2) dx?

In this case, u^2 is x^2 and a^2 is 25. Substitute u and a into the formula for arc tangent, giving (3/5)arc tangent(x/5) + C as the antiderivative.

Q: How do you integrate 8 / (x * sqrt(4x^2 - 1)) dx?

Recognize that this integrates using the formula for arc secant. Let u^2 be 4x^2, which means u = 2x. Substitute u and a into the formula, giving (8/2)arc secant(2x) + C = 4arc secant(2x) + C as the antiderivative.

Summary & Key Takeaways

• The video teaches three formulas for integrating functions that involve inverse trig functions: arc sine, arc tangent, and arc secant.

• It provides step-by-step examples of using these formulas to integrate various functions.

• The video emphasizes the importance of completing the square and performing appropriate substitutions to simplify the integrals.