Trig and U substitution together (part 2)

TL;DR

This video demonstrates how to evaluate an indefinite integral using trigonometric substitution.

Transcript

In the last video, in order to evaluate this indefinite integral, we first made the substitution that x is equal to 3 sine theta. And then this got us to an integral of this form. Then we were able to break up these sines and cosines and use a little bit of our trig identities. To get it into the form where we could do u substitution, we did anothe... Read More

Key Insights

• 😑 Trigonometric substitution is a powerful method for evaluating integrals involving square root of quadratic expressions.
• 😑 The process involves making appropriate substitutions, simplifying using trigonometric identities, and utilizing techniques to express variables in terms of each other.
• 😑 Right triangles and trigonometric identities can be used to convert expressions between different trigonometric functions.
• 🍉 Undoing the substitutions is necessary to obtain the final answer in terms of the original variable.

Questions & Answers

Q: What is the first step in evaluating the indefinite integral using trigonometric substitution?

The first step is to make a substitution by setting x equal to 3 sine theta.

Q: How is the integral transformed to be suitable for u-substitution?

A second substitution is made by setting u equal to cosine of theta, which rearranges the integral into a form that can be simplified using u-substitution.

Q: How can u be expressed in terms of x?

There are two techniques: using the trigonometric identity cosine theta = square root(1 - sine squared theta) or drawing a right triangle and applying the Pythagorean theorem to find u in terms of x.

Q: What is the final expression for the integral in terms of x?

The final expression is 243 times (1 - x squared over 9) to the power of 5/2 over 5, minus (1 - x squared over 9) to the power of 3/2 over 3, plus a constant.

Summary & Key Takeaways

• The video begins with making a substitution to evaluate an indefinite integral, followed by breaking up sines and cosines using trig identities.

• Another substitution is made to convert the integral into a form suitable for u-substitution.

• The process of undoing the substitutions is discussed, with the focus on finding a way to express u in terms of x.

• Two techniques, involving trigonometric identities and right triangles, are demonstrated for expressing u in terms of x.

• The final answer to the integral is obtained by substituting the expression for u back into the integral.