use trigonometric substitution, not power rule

TL;DR

Learn a special technique for integrating x squared without using the power rule.

Transcript

all right we'll be integrating x squared but  we are not going to be using the power rule of   any kind meaning that we are not going to bring  the power to the front and the minus one no no   power rule for derivative and also of course do  not add one to the power and then divide it by   the new power no power rule for integral as well  and to ma... Read More

Key Insights

• ✊ The video showcases an alternative approach to integration without using the power rule.
• ☺️ Substituting x with sine theta allows for the derivation of a trigonometric identity.
• 👨‍💼 The formula for sine of two angles is used to simplify the integral of sine theta times sine 2 theta.

Questions & Answers

Q: Why is the power rule not used for integrating x squared in this video?

The video aims to showcase an alternative method for integration that does not rely on the power rule. This allows viewers to see different approaches and broaden their understanding of integration techniques.

Q: What substitution is made in the video to simplify the integral of x squared?

The video suggests substituting x with sine theta, which helps in transforming the integral into an expression involving trigonometric functions. This allows for the derivation of a trigonometric identity that simplifies the integral further.

Q: How is the integral of sine theta times sine of 2 theta simplified?

The video uses a formula for sine of two angles (alpha and beta) to simplify the integral. By substituting alpha with theta and beta with 2 theta, the formula becomes sine theta times sine 2 theta = 1/2 cos(theta - 2theta) - 1/2 cos(theta + 2theta).

Q: How is the final solution obtained for the integral of x squared without the power rule?

After simplifying the integral using trigonometric identities, the video substitutes x back into the expression to obtain the final solution. The result is 1/3 x^3, without the need for the power rule.

Summary & Key Takeaways

• The video demonstrates a different method for integrating x squared without using the power rule.

• By substituting x with sine theta, the integral is transformed into an expression involving trigonometric identities.

• Using a formula for sine of two angles, the integral can be simplified further and solved without using the power rule.