Integration Techniques and Antiderivatives

This video tutorial covers the integration of functions leading to inverse trigonometric functions. It introduces three key formulas for AR sign, AR tan, and AR secant, and demonstrates their application through various examples. The tutorial also explores complex integration techniques, including completing the square and solving definite integrals.

Which formula is used to find the antiderivative of du / (a^2 - u^2)?

• Arctan(u/a) + C

• None of the above

• Arcsin(u/a) + C

• Arcsec(u/a) + C

Which inverse trigonometric function is used for the integral of du / (a^2 + u^2)?

• None of the above

• Arcsin

• Arctan

• Arcsec

In the example of integrating 3 over 25 + x^2, what is the value of 'a'?

• 1

• 3

• 5

• 25

In the example of integrating x / (x^2 + 1), what substitution is used for 'u'?

• u = x - 1

• u = x^2 - 1

• u = x^2 + 1

• u = x + 1

When completing the square for the expression x^2 - 4x + 7, what is the resulting perfect square trinomial?

• (x - 2)^2 + 3

• (x + 2)^2 + 3

• (x - 3)^2 + 4

• (x + 3)^2 + 4

In the problem involving e^(3x) / (9 + e^(6x)), what is the value of 'a'?

• e^(6x)

• e^(3x)

• 9

• 3

What substitution is made for 'x' in the integration of 8 over x * sqrt(4x^2 - 1)?

• x = 4u

• x = u/2

• x = 2u

• x = u^2

What is the antiderivative of 1/u?

• 1/u + C

• u^2/2 + C

• e^u + C

• ln|u| + C

In the definite integral problem, what is the new variable 'u' when x = 3?

• 0

• 6

• 3

• 9

For the definite integral problem, what is the final result of the integration?

• 1/5 Arcsec(3/5)

• 1/5 Arcsin(3/5)

• 1/5 Arctan(3/5)

• 1/5 Arctan(5/3)