How to Evaluate Definite Integrals Using Parts

TL;DR

Use integration by parts to evaluate the definite integral of x multiplied by cos(x) from 0 to π. Choose x as the function that simplifies upon differentiation and cos(x) which remains manageable when integrated, allowing you to apply the integration by parts formula effectively.

Transcript

• [Instructor] We're gonna do in this video is try to evaluate the definite integral from zero to pi of x cosine of x dx. Like always, pause this video and see if you can evaluate it yourself. Well when you immediately look at this, it's not obvious how you just straight up take the anti-derivative here and then evaluate that at pi and then subtrac... Read More

Key Insights

• 🥳 Integration by parts is a useful technique for evaluating definite integrals involving a product of functions.
• 🥳 Choosing the functions in integration by parts involves finding one that simplifies when its derivative is taken and another that doesn't get more complicated when its anti-derivative is found.
• ☺️ The anti-derivative of cosine of x is sine of x, while the derivative of x is one, which makes them suitable for integration by parts in this case.
• 😑 The integration by parts formula allows us to evaluate the definite integral by applying it and evaluating the resulting expression at the bounds.
• 🇦🇬 The arbitrary constants introduced in the anti-derivative calculations cancel out in definite integrals.

Questions & Answers

Q: What is integration by parts?

Integration by parts is a technique used to evaluate integrals that involve a product of functions. It involves finding two functions, one that simplifies when its derivative is taken and the other one that doesn't get more complicated when its anti-derivative is found.

Q: Why do we need to use integration by parts in this specific integral?

We need to use integration by parts because the definite integral involves a product of x and cosine of x. Taking the anti-derivative of x simplifies it, while the anti-derivative of cosine of x doesn't get more complicated.

Q: How is the integration by parts formula used to evaluate the definite integral?

The integration by parts formula states that the integral of f(x)g'(x) dx is equal to f(x)g(x) - ∫(f'(x)g(x)) dx. By substituting x for f(x) and sine of x for g(x), the definite integral can be evaluated by applying the formula and evaluating the resulting expression at the bounds.

Q: What is the value of the definite integral from zero to pi of x cosine of x?

The value of the definite integral is -2.

Summary & Key Takeaways

• The video explains how to evaluate the definite integral from zero to pi of x multiplied by the cosine of x using integration by parts.

• Integration by parts involves finding functions, one of which simplifies when its derivative is taken and the other one that doesn't get more complicated when its anti-derivative is found.

• By choosing x as the function that simplifies and cosine of x as the function that doesn't get more complicated, the definite integral is evaluated using the integration by parts formula.