Understanding the Indefinite Integral

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Questions and Answers

What is the indefinite integral of $2x$ with respect to $x$?

$2x^2 + C$
$x^2$
$2x + C$
$x^2 + C$ (correct)

What is the indefinite integral of $4$ with respect to $x$?

$4 + C$
$4x + C$ (correct)
$4x^2 + C$
$4x$

What is the indefinite integral of $3x^2$ with respect to $x$?

$x^2 + C$
$x^3 + C$ (correct)
$3x^2 + C$
$3x^3 + C$

What is the indefinite integral of $5e^x$ with respect to $x$?

$5e^x + C$ (D) Signup and view all the answers

What is the indefinite integral of $7 ext{sin}(x)$ with respect to $x$?

$-7 ext{cos}(x) + C$ (B) Signup and view all the answers

What is the indefinite integral of $rac{1}{x}$ with respect to $x$?

$ ext{ln}|x| + C$ (B) Signup and view all the answers

What is the indefinite integral of $rac{1}{ ext{cos}^2(x)}$ with respect to $x$?

$ ext{tan}(x) + C$ (D) Signup and view all the answers

What is the indefinite integral of $rac{1}{1+x^2}$ with respect to $x$?

$ ext{arctan}(x) + C$ (A) Signup and view all the answers

What is the indefinite integral of $rac{1}{ ext{sin}(x) ext{cos}(x)}$ with respect to $x$?

$ ext{ln}| ext{tan}(x)| + C$ (C) Signup and view all the answers

What is the indefinite integral of $rac{x+1}{ ext{x}^2+2}$ with respect to $x$?

$rac{ ext{ln}|(x^2+2)|}{2}+ ext{x}+C$ (B) Signup and view all the answers

Flashcards

What is an indefinite integral?

The indefinite integral of a function f(x) represents all its antiderivatives, denoted as ∫f(x) dx. It is a family of functions whose derivatives are f(x).

Indefinite integral of a constant.

The indefinite integral of a constant c is cx + C, where C is the constant of integration. This constant represents the unknown constant that disappears during differentiation.

Indefinite integral of a power function.

The indefinite integral of a power function x^n is (x^(n+1))/(n+1) + C, where n is any real number except -1, and C is the constant of integration. This rule is derived from the power rule of differentiation.

Indefinite integral of a sum of functions.

The indefinite integral of a sum of functions is the sum of the indefinite integrals of each function. This is a result of the linearity property of differentiation.