Antiderivatives and Integration Formulas

The video tutorial explains how to find the antiderivative of functions by determining their indefinite integrals. It covers three examples, each demonstrating different integration techniques and formulas. The first example involves factoring out constants and using the inverse tangent formula. The second example uses inverse secant, requiring factorization of the denominator. The third example involves simplifying the integrand to match the inverse sine formula. Each example is explained step-by-step, highlighting the importance of recognizing integration formulas and simplifying expressions.

What is the primary goal when finding an antiderivative?

• To find the derivative of a function

• To determine the definite integral

• To determine the indefinite integral

• To solve a differential equation

Which integration formula is used for the first function?

• Inverse cosine

• Inverse secant

• Inverse tangent

• Inverse sine

How is the constant factor handled in the first function's antiderivative?

• It is multiplied by the variable

• It is ignored

• It is added to the denominator

• It is factored out

What is the antiderivative of the first function?

• Seven times inverse cosine of x plus c

• Seven times inverse secant of x plus c

• Seven times inverse tangent of x plus c

• Seven times inverse sine of x plus c

What additional factor is present in the denominator of the second function?

• A factor of four

• A factor of five

• A factor of three

• A factor of two

Which integration formula is applied to the second function?

• Inverse tangent

• Inverse cosine

• Inverse sine

• Inverse secant

What is the antiderivative of the second function?

• 5 halves times inverse secant of the absolute value of x plus c

• 5 halves times inverse cosine of x plus c

• 5 halves times inverse sine of x plus c

• 5 halves times inverse tangent of x plus c

What transformation is necessary for the third function's integrand?

• Differentiating the function

• Completing the square

• Factoring out a constant

• Changing the variable

Which integration formula is used for the third function?

• Inverse sine

• Inverse tangent

• Inverse secant

• Inverse cosine

What is the antiderivative of the third function?

• One half inverse secant x plus c

• One half inverse cosine x plus c

• One half inverse tangent x plus c

• One half inverse sine x plus c