Antiderivatives Review

What is the antiderivative of @@\frac{1}{x}@@?

• @@rac{1}{2}x^2 + C@@

• @@rac{1}{x} + C@@

• @@rac{1}{x^2} + C@@

• @@ ext{ln}|x| + C@@

What is the antiderivative of @@7x^5 - 4x^3 + 8@@?

• @@rac{7}{6}x^6 - x^4 + 8x + C@@

• @@rac{7}{6}x^6 - 4x^4 + 8x + C@@

• @@rac{7}{6}x^6 - 4x^3 + 8x + C@@

• @@rac{7}{6}x^6 - x^4 + 4x + C@@

Determine the antiderivative of @@x^2@@.

• @@rac{1}{3}x^3 + C@@

• @@rac{1}{2}x^3 + C@@

• @@x^3 + C@@

• @@rac{1}{4}x^3 + C@@

What is the antiderivative of a constant k?

• kx + C

• k/x + C

• k^2x + C

• k + C

State the integral of a sum of functions.

• The integral of a sum of functions is the product of their integrals: @@rac{1}{2} imes rac{1}{3}@@ + C

• The integral of a sum of functions is the sum of their integrals: @@rac{1}{2} + rac{1}{3} + C@@

• The integral of a sum of functions is the difference of their integrals: @@rac{1}{2} - rac{1}{3} + C@@

• The integral of a sum of functions is the sum of their integrals: @@ extstyle rac{1}{2} + rac{1}{3} + C@@.

Define an antiderivative.

• An antiderivative of a function f(x) is a function F(x) such that F'(x) = f(x).

• An antiderivative is the integral of a function over a specific interval.

• An antiderivative is a function that has no derivatives.

• An antiderivative is a function that is always increasing.

How do you find the antiderivative of a polynomial function?

• Use the power rule for each term: increase the exponent by 1 and divide by the new exponent.

• Multiply each term by the original exponent and decrease the exponent by 1.

• Add all the coefficients of the polynomial and divide by the number of terms.

• Integrate the polynomial by applying the product rule for differentiation.

• @@rac{1}{x} + C@@

• @@rac{x^2}{2} + C@@

• @@rac{x^{-1}}{-1} + C@@

• @@ ext{ln}|x| + C@@

Explain the constant of integration (C).

• The constant of integration (C) represents an infinite number of possible vertical shifts of the antiderivative.

• The constant of integration (C) is always equal to zero in all cases.

• The constant of integration (C) is used to denote the slope of the function.

• The constant of integration (C) is irrelevant in calculus.