Integral Calculus: Indefinite Integration

Indefinite Integration

• Derivative of x with respect to x is nx; integral of x dx equals x²/2 + c.
• Derivative of sin x is cos x; integral of cos x dx equals sin x + c.
• Derivative of cos x is -sin x; integral of sin x dx equals -cos x + c.
• Derivative of tan x is sec² x; integral of sec² x dx equals tan x + c.
• Derivative of cosec x is -cosec x cot x; integral of cosec x cot x dx equals -cosec x + c.
• Derivative of sec x is sec x tan x; integral of sec x tan x dx equals sec x + c.
• Derivative of cot x is -cosec² x; integral of cosec² x dx equals -cot x + c.
• Exponential derivative, d/dx(e^x) = e^x; integral, ∫e^x dx equals e^x + c.
• Derivative of ln(x) is 1/x; integral, ∫(a^x) dx = (a^x/ln a) + c.
• Integral of 1/√(1-x²) dx is sin⁻¹(x) + c; integral of -1/√(1-x²) dx is cos⁻¹(x) + c.

Relations and Functions

• Function f(x) = x behaves as follows:
  • Positive for x > 0
  • Zero for x = 0
  • Negative for x < 0
• Assume f(x) = -1 if x < 0.
• Analyze injectivity, surjectivity, and bijection for functions:
  • f(x) = x is bijective (both injective and onto).
  • f(x) = 3x is also bijective.

Integrations

• Integral of a constant multiplied by dx equals x + c.
• Integral of the sum of two functions is the sum of their integrals.
• Integral of k times f(x) dx is k times the integral of f(x) dx.

Substitution Method

• Formula for ∫(ax + b)ⁿ dx: (1/a) * (ax + b)^(n+1)/(n+1) + c.
• ∫sec²(ax + b)dx = (1/a)tan(ax + b) + c.
• ∫sin(ax + b) dx = -(1/a)cos(ax + b) + c.
• ∫cos(ax + b) dx = (1/a)sin(ax + b) + c.
• ∫cosec²(ax + b)dx = -(1/a)cot(ax + b) + c.
• ∫sec(ax + b)tan(ax + b) dx = (1/a)sec(ax + b) + c.
• ∫cosec(ax + b)cot(ax + b)dx = -(1/a)cosec(ax + b) + c.

Integration Formulas

• ∫(1/(ax+b)) dx = (1/a)ln|ax+b| + c.
• ∫e^(ax+b) dx = (1/a)e^(ax+b) + c.
• ∫(ax+b) dx = (a/2)x² + bx + c.
• ∫tan x dx = ln|sec x| + c.
• ∫cot x dx = ln|sin x| + c.
• ∫cosec x dx = ln|cosec x - cot x| + c.
• ∫sec x dx = ln|sec x + tan x| + c.
• For f'(x)f(x) integral, use (1/a)ln|f(x)| + c for linear functions.

Methods of Differentiation

• Derivative of sec x is sec x tan x.
• Derivative of cot x is -cosec² x.
• Derivative of tan x is sec² x.
• Derivative of cosec x is -cosec x cot x.
• Derivative of ln x is 1/x.
• Derivative of e^x is e^x.
• Derivative of a^x is a^x ln a.
• Derivative of arccot x is -1/(1 + x²).
• Derivative of arcsec x is 1/(|x|√(x² - 1)).
• Derivative of arccosec x is -1/(|x|√(x² - 1)).

Properties of Derivatives

• The derivative of a sum d/dx[f(x) + g(x)] = d/dx f(x) + d/dx g(x).
• Scaling property: d/dx[kf(x)] = k d/dx f(x).
• Product rule: d/dx(uv) = u(d/dx v) + v(d/dx u).
• Quotient rule: d/dx(u/v) = (v(d/dx u) - u(d/dx v))/v².
• Chain rule: d/dx[f(g(x))] = f'(g(x))(d/dx g(x)).