Integration Techniques and Functions

These flashcards cover key integration techniques and functions related to basic functions, trigonometric functions, and their respective integrals and derivatives.

Reverse Product Rule

F dS = FS - S dF + C (Integration by Parts)

Reverse Chain Rule in Integration

Also known as u-substitution.

Basic Function Derivative

The derivative of -x^n is -nx^(n-1).

Integral of x^P

∫ x^P dx = (x^(P+1))/(P+1) + C

Integral of √x

∫ √x dx = (2/3)x^(3/2) + C.

Derivative of ln(a)

d(ln a)/da = 1/a.

Integral of sin x

∫ sin x dx = -cos x + C.

Derivative of cos x

d(cos x)/dx = -sin x.

Integral of sec^2 x

∫ sec^2 x dx = tan x + C.

Integral of tan x

∫ tan x dx = ln |sec x| + C.

Integral of cot x

∫ cot x dx = ln |sin x| + C.

Integral of sec x tan x

∫ sec x tan x dx = sec x + C.

Integral of csc x cot x

∫ csc x cot x dx = -csc x + C.

Antiderivative of sec^2 x

∫ sec^2 x dx = tan x + C.

Antiderivative of csc^2 x

∫ csc^2 x dx = -cot x + C.

Integral of arcsin

∫ arcsin(x) dx = x * arcsin(x) + sqrt(1-x^2) + C.

Integral involving arctangent

∫ arctan(x) dx = x * arctan(x) - (1/2)ln(1+x^2) + C.

Integral of arcsec

∫ arcsec(x) dx = x * arcsec(x) - ln|x + sqrt(x^2-1)| + C.