How to Solve the Integral of sin^2x*cos^2x

TL;DR

To solve the integral of sin²x*cos²x, rewrite it as (1/4) * ∫sin(2x)² dx using the double angle identity. Then apply the power reduction formula, integrate, and simplify to obtain the final answer, which is (1/8)x - (1/32)sin(4x) + C.

Transcript

okay this interposed for the calculus 2 students and this is the pretty popular integral that you need to know how to solve in your calculus 2 class the integral of sine square x times cosine squared X first of all you may be thinking let's just convert the sine square X or maybe the other one but let's to care the sine square X let's change that t... Read More

Key Insights

• ❎ Converting sine squared X or cosine squared X does not simplify the integral.
• 😒 The use of u-substitution is not suitable for this particular integral because of the mismatch in derivatives.
• 👻 Rewriting the integral in terms of sine X times cosine X squared allows the application of the double angle identity.
• ✊ The power reduction formula for sine squared X can be used to integrate the function.

Questions & Answers

Q: Why shouldn't we convert sine squared X or cosine squared X in the integral?

Converting sine squared X or cosine squared X in the integral will not simplify the problem. It is more efficient to take a different approach.

Q: Can we use u-substitution to solve the integral?

No, u-substitution is not applicable here because of the mismatch between the derivative of sine X and the cosine squared X present in the integral.

Q: What does rewriting the integral as the integral of sine X times cosine X squared accomplish?

Rewriting the integral allows us to apply the double angle identity for sine X times cosine X, simplifying the problem.

Q: How do we integrate sine squared X times cosine squared X after rewriting the integral?

After rewriting, we can use the power reduction formula for sine squared, which yields 1/2 times 1 minus cosine of 2X, to integrate the function.

Summary & Key Takeaways

• The video explains that converting sine squared X or cosine squared X will not simplify the integral.

• It advises against using u-substitution because the derivatives of the trigonometric functions involved do not cancel each other out.

• Instead, the video suggests rewriting the integral as the integral of sine X times cosine X, both raised to the second power.