Integrals and Trigonometric Identities

This video tutorial explains how to integrate the function x squared over x squared plus 1. The process involves adding and subtracting 1 in the numerator, splitting the integral into two parts, and using trigonometric substitution. The tutorial demonstrates the use of Pythagorean identities and concludes with the final solution: x minus the arc tangent of x plus a constant.

What is the first step taken to simplify the integral of x squared over x squared plus 1?

• Subtract 1 from the denominator

• Add 1 and -1 to the numerator

• Multiply the numerator by 2

• Divide the numerator by 2

Why is it valid to add 1 and -1 to the numerator in the initial step?

• It makes the integral zero

• It does not change the value of the numerator

• It simplifies the denominator

• It changes the value of the numerator

How is the expression x squared plus 1 divided by itself simplified?

• It becomes zero

• It becomes negative one

• It remains the same

• It becomes one

What happens to the negative sign in the integral of -1 over x squared plus 1?

• It is multiplied by the denominator

• It is added to the numerator

• It is moved to the front of the integral

• It is ignored

What substitution is used to simplify the integral of 1 over x squared plus 1?

• x = sec(u)

• x = sin(u)

• x = cos(u)

• x = tan(u)

What is the derivative of tangent used in the substitution?

• sin squared u

• cos squared u

• sec squared u

• tan squared u

Which trigonometric identity is used to replace tan squared plus 1?

• tan squared = sec squared - 1

• 1 + cos squared = sec squared

• 1 + tan squared = sec squared

• sin squared + cos squared = 1

What is the integral of 1 with respect to u?

• u + C

• sec(u) + C

• x + C

• tan(u) + C

What is the final expression for the integral of x squared over x squared plus 1?

• x + arctan(x) + C

• x - arctan(x) + C

• x + C

• arctan(x) - x + C

What is the inverse function used to find u in terms of x?

• arccos(x)

• arcsec(x)

• arcsin(x)

• arctan(x)