Unit 6 Study Guide: Antiderivatives & Integration Methods

Integration

The inverse operation of differentiation that calculates accumulation such as total displacement or area under a curve.

Antiderivative

A function F(x) is an antiderivative of f(x) if F'(x) = f(x) for all x in an interval.

General Antiderivative

The family of functions represented as F(x) + C, where C is an arbitrary constant.

Indefinite Integral

The notation used to express all antiderivatives of a function, written as ∫f(x)dx = F(x) + C.

Power Rule for Integration

The rule for reversing the Power Rule for differentiation: ∫x^n dx = x^(n+1)/(n+1) + C, where n ≠ -1.

Special Case of Power Rule

For n = -1, the integral is ∫(1/x)dx = ln|x| + C.

Trigonometric Integrals

Standard integration formulas for trigonometric functions that include results for sin(x) and cos(x).

Linearity of Integration

Integration is linear, allowing for the splitting of sums and pulling out constants.

Integration by Substitution

A method to reverse the Chain Rule used for integrals of composite functions.

Step 1 of Integration by Substitution

Choose u = g(x), where g(x) is a function whose derivative is present.

Polynomial Long Division

A technique used to simplify a rational function if the degree of the numerator is greater than or equal to the degree of the denominator.

Completing the Square

A method to rewrite a quadratic expression to apply integration techniques.

Integral of sin(x)

The indefinite integral ∫sin(x)dx = -cos(x) + C.

Integral of cos(x)

The indefinite integral ∫cos(x)dx = sin(x) + C.

Integral of e^x

The indefinite integral ∫e^xdx = e^x + C.

Common mistake: Omission of +C

Forgetting to add the constant of integration when performing indefinite integrals.

Common mistake: Variable confusion in u-substitution

Replacing parts of the function with u but leaving dx instead of converting to du during integration.

Common mistake: Incomplete chain rule reversal

Failing to account for constant multipliers when substituting variables in integration.

Common mistake: Log rule misapplication

Incorrectly applying the power rule to integrals involving algebraic fractions or rational functions.

Integral of sec^2(x)

The indefinite integral ∫sec^2(x)dx = tan(x) + C.

Integral of sec(x)tan(x)

The indefinite integral ∫sec(x)tan(x)dx = sec(x) + C.

Integral of csc^2(x)

The indefinite integral ∫csc^2(x)dx = -cot(x) + C.

Integral of csc(x)cot(x)

The indefinite integral ∫csc(x)cot(x)dx = -csc(x) + C.

Integral of a^x

The indefinite integral ∫a^xdx = (a^x)/(ln a) + C.

Integral of sec x

The indefinite integral ∫sec(x)dx is a standard result not covered in elementary rules.

Integral form for u^2 + a^2

∫(1/(u^2 + a^2))dx = (1/a)arctan(u/a) + C.

Integral involving a quadratic denominator

Often requires completing the square before using formulas for inverse trigonometric integrals.

Substitution for Composite Functions

If integrating ∫f(g(x))g'(x)dx, substitute u = g(x) and adjust for du.