Integration

If F'(x) = f(x)

If F'(x) = f(x), then F(x) is called the antiderivative of f(x). Integration is the process of finding antiderivatives. The symbol ∫f(x)dx represents

Key formulas include

Key formulas include: ∫x^n dx = x^(n+1)/(n+1) + C (n≠-1), ∫1/x dx = log|x| + C, ∫sin x dx = -cos x + C, ∫cos x dx = sin x + C, ∫sec²x dx = tan x + C,

This method transforms a complex integral

This method transforms a complex integral into a simpler form by substituting a new variable. For functions of the form f(ax+b), substitute u = ax+b,

Based on the product rule

Based on the product rule of differentiation, the formula is ∫u dv = uv - ∫v du. Choose 'u' as the function that simplifies upon differentiation, and

When the numerator is the derivative

When the numerator is the derivative of the denominator, the integral is simply log|f(x)| + C. Example: ∫(2x+3)/(x²+3x+1) dx = log|x²+3x+1| + C becaus