Integral of x*sin(x^2)

TL;DR

Learn how to integrate x sin(x^2) using u-substitution and trigonometric identities.

Transcript

in this example we have to integrate x times the sine of x squared solution so we'll start by making a u substitution we're going to let u be the inside function so in this case our inside function is going to be this piece here so we'll let u be equal to x squared now we're going to take the derivative of both sides of this equation and we're goin... Read More

Key Insights

• 🗞️ U-substitution is a powerful technique in calculus to simplify integrals by introducing a new variable.
• 😑 Adjusting the expression to match the integral form is crucial for successful integration.
• ❓ Utilizing trigonometric identities like sin(u) = -cos(u) can simplify the integration process significantly.

Questions & Answers

Q: What is the first step in integrating x sin(x^2)?

The first step is to apply u-substitution by letting u be the inside function (x^2) and taking its derivative to simplify the expression.

Q: How do you adjust the expression to match the integral form?

To match the integral form, adjust for the 2x term by dividing by two to make it dx/2, simplifying the expression to fit the integration.

Q: What trigonometric identity is used to integrate sin(x^2)?

The trigonometric identity used is the integral of sin(u) which results in -cos(u) + C, where u is equal to x squared.

Q: Why is it necessary to go back to the original variable after integration?

Going back to the original variable is crucial to ensure the final answer is in terms of x and not u, maintaining consistency in the integration process.

Summary & Key Takeaways

• The process begins with u-substitution, letting u be x squared and taking its derivative.

• Simplify the expression to match the form of the integral by adjusting for the 2x term.

• Integrate the simplified expression using trigonometric identities and go back to the original variable.