How to Integrate sin(x)cos(x) Using Three Techniques

TL;DR

To integrate sin(x)cos(x), you can use three methods: 1) Apply u-substitution with u = sin(x) to find the integral as 1/2 sin²(x) + C. 2) Use u = cos(x) to achieve -1/2 cos²(x) + C. 3) Utilize the double angle formula to rewrite the integral as -1/4 cos(2x) + C.

Transcript

in this video we're going to focus on finding the integral of sine x cosine x now there's three ways in which we can get the answer the first technique involves u substitution and we're going to replace u with sine x d u the derivative of sine is going to be cosine x dx and so let's substitute sine with the u variable and let's replace cosine x dx ... Read More

Key Insights

• 😄 U substitution is a useful technique for simplifying integrals involving trigonometric functions.
• 💄 Making u equal to a trigonometric function can provide an alternative approach to finding the integral.
• 💁 The double angle formula for sine 2x can be utilized to rewrite the integral in a different form.
• 👻 The three different forms of the double angle formula for cosine 2x allow for multiple representations of the integral.

Questions & Answers

Q: What are the three methods for finding the integral of sine x cosine x?

The three methods are u substitution, making u equal to cosine x, and using the double angle formula for sine 2x.

Q: How does the u substitution method work?

In the u substitution method, you replace sine x with the variable u and cosine x dx with du. The integral becomes the integral of u du.

Q: How does the second method of making u equal to cosine x work?

By making u equal to cosine x, you can solve for dx by taking the derivative of cosine x. The integral becomes the integral of sine x u du divided by negative sine.

Q: What is the third method involving the double angle formula for sine 2x?

The third method replaces sine x cosine x with one-half sine 2x in the integral. You then find the antiderivative of sine 2x and divide it by 2.

Summary & Key Takeaways

• The first technique involves using u substitution by replacing sine x with the variable u and cosine x dx with du.

• The second technique involves making u equal to cosine x and solving for dx by taking the derivative of cosine.

• The third technique uses the double angle formula for sine 2x to replace sine x cosine x in the integral.