How to Integrate the Square Root of sin^2(x)?

TL;DR

To integrate sqrt(sin^2(x)), recognize that it simplifies to |sin(x)|, leading to the integral of |sin(x)| dx. The presenter attempts various methods, including integration by parts and substitution, but struggles to find a clean solution matching Wolfram Alpha's result and invites viewers for suggestions.

Transcript

okay today I have a simple integral that I cannot  solve the integral of square root of sin Square X   and I know this right here is just the same as  the integral of absolute value of sinx and of   course we can just look at where is sinx negative  and then just negate the result pretty much do it   like piecewise right but I don't want to do that... Read More

Key Insights

• 🥳 The presenter explores different strategies such as integration by parts, rationalizing the denominator, and substitution.
• 🥅 The goal is to find a solution to the integral that matches the answer obtained from Wolfram Alpha.
• ❓ Despite multiple attempts, the presenter is unable to reach a solution and seeks assistance.

Questions & Answers

Q: Can you take the square root of sin^2(x)?

Yes. The presenter states that the square root of sin^2(x) is just the absolute value of sin(x). Because of that, the integral of sqrt(sin^2(x)) is the same as the integral of |sin x| dx.

Q: What integral is the video trying to solve?

He is trying to integrate the square root of sin^2(x), which equals the integral of |sin x|. His stated goal is to reach the same closed-form answer he obtained from Wolfram Alpha rather than settle for a piecewise result.

Q: Why not just solve it piecewise?

The presenter acknowledges you could look at where sin(x) is negative and negate the result, handling it piecewise. He deliberately avoids that because he wants a single clean expression that matches the Wolfram Alpha answer instead of a case-by-case one.

Q: What methods does he attempt in the video?

He verifies a candidate answer by differentiation (using the product and chain rules), tries integration by parts with the DI method, rationalizes the denominator to turn the bottom into sin^2(x), and uses substitution. None of these get him fully to the target answer.

Q: How does he avoid ending up with an absolute value?

When differentiating to check his candidate, he keeps the square root of sin^2(x) intact instead of cancelling it to |sin x|. After rationalizing the denominator, the terms factor into sqrt(sin^2 x) times (cosecant^2 x minus cotangent^2 x), and since that bracket equals 1, he lands back on sin^2(x) without introducing absolute values.

Q: What is the integral of sqrt(x^2), and how does he find it?

He shows that the integral of sqrt(x^2) dx equals one-half times x times sqrt(x^2), plus C, which is also the integral of |x|. He gets it with integration by parts (the DI method), rationalizing the denominator, then adding the integral back to both sides and dividing by two.

Q: Does the presenter actually solve the integral of sqrt(sin^2(x))?

No. Despite trying for many weeks and going through integration by parts, rationalizing the denominator, and substitution, he cannot simplify it to the Wolfram Alpha answer. He shares his work and explicitly asks viewers for alternative approaches.

Q: Why does he check his work by differentiation?

To test whether a proposed antiderivative is correct, he differentiates it and simplifies. Using the product rule, the chain rule, and the identity that cosecant^2 x minus cotangent^2 x equals 1, he confirms the derivative reduces back to the original integrand.

Summary & Key Takeaways

• The video presenter tries to find a solution to the integral of the square root of sin squared x, aiming to reach the answer obtained from Wolfram Alpha.

• Multiple methods are attempted, including using integration by parts, rationalizing the denominator, and substitution.

• Despite several attempts, the presenter is unable to simplify the integral and seeks help for alternative approaches.