Integral of x^2/(x^2+1)

TL;DR

Learn how to integrate the expression x squared over x squared plus 1 using trigonometric substitution.

Transcript

in this lesson we're going to talk about integrating x squared over x squared plus 1 dx so what do you think we need to do in this problem what we're going to do is we're going to add 1 and negative 1 to the numerator the reason why we can do this is because the value of the numerator hasn't changed it's still equal to x squared x squared plus 1 mi... Read More

Key Insights

• ☺️ In this math lesson, the integral of x squared over x squared plus 1 is solved using trigonometric substitution.
• 👻 Adding 1 and -1 to the numerator does not change its value and allows for simplification.
• 😑 By breaking the expression into two fractions, they can be integrated separately.
• ☺️ Trigonometric substitution involves setting x equal to tangent of u.
• ❎ The integral of 1dx is x, and the integral of tan squared plus 1 is secant squared.
• 😑 Canceling out the common factors simplifies the integration expression.
• ☺️ The final answer to the integration problem is x minus arc tangent of x plus a constant.

Questions & Answers

Q: How do you simplify the expression x squared plus 1 divided by x squared plus 1?

The expression x squared plus 1 divided by x squared plus 1 simplifies to 1 because any number divided by itself is equal to 1.

Q: What substitution is used in this integration problem?

Trigonometric substitution is used in this integration problem. x is set equal to tangent of u, and dx is replaced with secant squared du.

Q: What is the integral of 1dx?

The integral of 1dx is simply x, as the derivative of x is 1.

Q: How can the expression tan squared plus 1 be replaced with secant squared?

The expression tan squared plus 1 can be replaced with secant squared using the Pythagorean identity tan squared plus 1 equals secant squared.

Summary & Key Takeaways

• To integrate x squared over x squared plus 1, you add 1 and -1 to the numerator to maintain its equality.

• Break the expression into two fractions: x squared plus 1 divided by the denominator and -1 divided by the denominator.

• Use trigonometric substitution by setting x equal to tangent of u and replacing dx with secant squared du.