problem 2 based on Differentiation under Integral Sign one parameter

TL;DR

Learn the concept of differentiation under the integral sign and solve a numerical problem using the five-step method.

Transcript

hello students so in this video we are gonna learn the concept of differentiation under integral sign and we are gonna solve the numerical based on the same concept so here we have been given a question that is we have to show that integration from 0 to pi log of 1 plus alpha cos x whole upon x dx as equal to pi sine inverse a now the question is h... Read More

Key Insights

• 🤘 Differentiation under the integral sign is a concept used to solve integrals with variables and parameters.
• 🤘 The five-step method helps in solving integrals using differentiation under the integral sign.
• 🙃 The method involves applying duis rule, integrating both sides, and substituting values to find the constant of integration.
• 🈸 The example provided demonstrates the application of the five-step method to solve a specific integral problem.
• ☺️ The solution is derived using the method of integral dx upon a plus b cos x and the formula of tan inverse.

Questions & Answers

Q: How do you identify if a problem belongs to differentiation under the integral sign?

A problem belongs to differentiation under the integral sign if the integral contains a variable with a parameter. In this case, the variable is x and the parameter is a.

Q: What are the five steps to solve integrals using differentiation under the integral sign?

The five steps are: 1) Consider the given integral as i of alpha; 2) Apply duis rule and find di by da; 3) Integrate both sides to find i of alpha; 4) Substitute a specific value of alpha to find the constant of integration; 5) Substitute the value of the constant to find the final value of the integral.

Q: How do you solve the integral in the given example?

In the example, the method of integral dx upon a plus b cos x is used. The substitution t = tan(x/2) is made, and the limits are determined for the new variable t. The integral is then simplified and evaluated using the formula of tan inverse.

Q: What is the final value of the integral in the example?

The final value of the integral is pi times the sine inverse of a, which was proven using the rule of duis.

Summary & Key Takeaways

• The video introduces the concept of differentiation under the integral sign.

• The five-step method for solving integrals is explained.

• A specific numerical problem is solved using the differentiation under the integral sign rule.