Indefinite Integral
- Definition: The indefinite integral of a function f(x) is denoted by ∫f(x)dx.
- It represents the family of antiderivatives of f(x).
- The indefinite integral finds the area under the curve of a function.
- The indefinite integral of a constant C is Cx + K, where K is the constant of integration.
- The integral of the sum of two functions is the sum of their integrals.
- The integral of a constant multiplied by a function is equal to the constant multiplied by the integral of the function.
- The integral of the product of a constant and a function is equal to the constant multiplied by the integral of the function.
- Basic integration formulas:
- ∫1dx = x + C
- ∫x^ndx = (x^(n+1))/(n+1) + C, (n ≠ -1)
- ∫e^xdx = e^x + C
- ∫a^xdx = (a^x)/(ln(a)) + C, (a > 0, a ≠ 1)
- The integral of a power of the natural logarithm:
- ∫ln(x)dx = xln(x) - x + C
- The integral of a rational function:
- ∫(1/x)dx = ln|x| + C
- The integral of a power of a trigonometric function:
- ∫sin^nx dx = (-1/n)(sin^(n-1)x)(cosx) + (n-1)/n ∫sin^(n-2)x dx, (n ≠ 1)
- Example:
- ∫sin^3x dx = (-1/3)(sin^2x)(cosx) + (2/3) ∫sinx dx
- Simplifying further, we get: -1/3 sin^2x cosx + 2/3 ∫sinx dx
- The integral of a power of a secant function:
- ∫sec^nx dx = (1/n-1) (sec^(n-2)x)(tanx) + (n-2)/(n-1) ∫sec^(n-2)x dx, (n ≠ 1)
- Example:
- ∫sec^3x dx = (1/2) (secx)(tanx) + (1/2) ∫secx dx
- The integral of a power of a cosecant function:
- ∫csc^nx dx = (-1/n-1) (csc^(n-2)x)(cotx) + (n-2)/(n-1) ∫csc^(n-2)x dx, (n ≠ 1)
- Example:
- ∫csc^3x dx = (-1/2) (cscx)(cotx) + (1/2) ∫cscx dx
- Integration by parts formula:
- ∫udv = uv - ∫vdu
- The integral of the product of two functions is equal to the product of the first function and the integral of the second function, minus the integral of the derivative of the first function multiplied by the integral of the second function.
- Example:
- ∫xsin(x) dx = -xcos(x) + ∫cos(x) dx
- Integration using substitution:
- Replace a variable with a new variable.
- Differentiate the new variable to find du.
- Evaluate the integral using the new variable and du.
- If necessary, substitute the original variable back in at the end.
- Example:
- ∫3x^2√(x^3 + 1) dx
- Let u = x^3 + 1
- Du = 3x^2 dx
- Substituting, the integral becomes: ∫√u du
- Integration of trigonometric functions:
- ∫sin(ax) dx = (-1/a) cos(ax) + C
- ∫cos(ax) dx = (1/a) sin(ax) + C
- Example:
- ∫sin(2x) dx = (-1/2) cos(2x) + C
- ∫cos(3x) dx = (1/3) sin(3x) + C
- Integration of exponential functions:
- ∫e^(ax) dx = (1/a) e^(ax) + C
- Example:
- ∫e^(3x) dx = (1/3) e^(3x) + C
- Integration of expressions involving logarithmic functions:
- ∫ln(ax) dx = x ln(ax) - x + C, a > 0, a ≠ 1
- Example:
- ∫ln(2x) dx = x ln(2x) - x + C
- Integration of inverse trigonometric functions:
- ∫arcsin(ax) dx = x arcsin(ax) + (1/a) √(1 - (ax)^2) + C
- ∫arccos(ax) dx = x arccos(ax) - (1/a) √(1 - (ax)^2) + C
- Example:
- ∫arcsin(2x) dx = x arcsin(2x) + (1/2) √(1 - (2x)^2) + C
- Integration using partial fractions:
- Express a rational function as the sum of simpler fractions.
- Determine the unknown coefficients.
- Integrate each fraction separately.
- Example:
- ∫(x^2 + 3x + 2)/((x - 1)(x + 2)) dx
- Rewrite as: ∫(A/(x - 1) + B/(x + 2)) dx
- Determine A and B, and then integrate each term separately.
- Integration using trigonometric identities:
- ∫sin^2(x) dx = (1/2) (x - sin(x)cos(x)) + C
- ∫cos^2(x) dx = (1/2) (x + sin(x)cos(x)) + C
- Example:
- ∫sin^2(2x) dx = (1/2) (x - sin(2x)cos(2x)) + C
- Integration using logarithmic properties:
- ∫(1/x) dx = ln|x| + C
- ∫(1/(x+a)) dx = ln|x+a| + C
- Example:
- ∫(1/(x+2)) dx = ln|x+2| + C
- Integration of trigonometric functions with odd exponents:
- ∫sin^(2n+1)(x) dx = -1/(2n+1) * cos(sin^(2n)x) + C
- ∫cos^(2n+1)(x) dx = 1/(2n+1) * sin(cos^(2n)x) + C
- Example:
- ∫sin^3(x) dx = -1/3 * cos^3(x) + C
- Integration of inverse hyperbolic functions:
- ∫arcsinh(x) dx = x*arcsinh(x) - √(x^2 + 1) + C
- ∫arccosh(x) dx = x*arccosh(x) - √(x^2 - 1) + C
- Example:
- ∫arcsinh(2x) dx = x*arcsinh(2x) - √(4x^2 + 1) + C
- Integration of hyperbolic functions:
- ∫sinh(x) dx = cosh(x) + C
- ∫cosh(x) dx = sinh(x) + C
- Example:
- ∫sinh(3x) dx = cosh(3x) + C
- Integration of exponential functions:
- ∫e^(ax)sin(bx) dx = (ae^(ax)sin(bx) - be^(ax)cos(bx))/(a^2 + b^2) + C
- ∫e^(ax)cos(bx) dx = (ae^(ax)cos(bx) + be^(ax)sin(bx))/(a^2 + b^2) + C
- Example:
- ∫e^(2x)sin(x) dx = (2e^(2x)sin(x) - e^(2x)cos(x))/5 + C
- Integration using trigonometric substitution:
- Substitute the variable with a trigonometric function.
- Rewrite the expression using trigonometric identities.
- Evaluate the integral using the new substitution.
- Example:
- ∫(x^2√(25 - x^2)) dx
- Let x = 5sin(θ)
- Rewrite the expression using trigonometric identities.
- Evaluate the integral using the new substitution.
- Integration of rational functions using partial fractions:
- Factorize the denominator into linear and irreducible quadratic factors.
- Express the rational function as the sum of partial fractions.
- Determine the unknown coefficients using suitable operations.
- Integrate each fraction separately.
- Example:
- ∫(4x^2 + 3x + 1)/(x^2 - 1)(x + 2) dx
- Factorize the denominator: (x - 1)(x + 1)(x + 2)
- Express as partial fractions and determine the coefficients.
- Integrate each fraction separately.
- Integration of improper rational functions:
- Identify the type of improper rational function.
- Apply the appropriate method of integration.
- Evaluate the integral.
- Example:
- ∫(x^3 + 2x + 1)/(x^2 - 4) dx
- Improper rational function: degree of numerator ≥ degree of denominator
- Apply long division or partial fractions to simplify the expression.
- Evaluate the integral.
- Definite integral:
- The definite integral finds the area between the curve and the x-axis over a specified interval.
- The notation for definite integral:
- ∫[a, b] f(x) dx
- Properties of definite integrals:
- ∫[a, b] f(x) dx = -∫[b, a] f(x) dx (changing the limits changes the sign)
- ∫[a, a] f(x) dx = 0 (the area between the same point is zero)
- Fundamental Theorem of Calculus:
- Part 1: If f(x) is continuous on [a, b], then the function F(x) = ∫[a, x] f(t) dt is differentiable and F’(x) = f(x).
- Fundamental Theorem of Calculus:
- Part 2: If f(x) is continuous on [a, b], then ∫[a, b] f(x) dx = F(b) - F(a), where F(x) is the antiderivative of f(x).
- Example:
- ∫[2, 5] (3x^2 - 2) dx = F(5) - F(2), where F(x) is the antiderivative of (3x^2 - 2).
Indefinite Integral Definition: The indefinite integral of a function f(x) is denoted by ∫f(x)dx. It represents the family of antiderivatives of f(x). The indefinite integral finds the area under the curve of a function. The indefinite integral of a constant C is Cx + K, where K is the constant of integration. The integral of the sum of two functions is the sum of their integrals. The integral of a constant multiplied by a function is equal to the constant multiplied by the integral of the function. The integral of the product of a constant and a function is equal to the constant multiplied by the integral of the function. Basic integration formulas: ∫1dx = x + C ∫x^ndx = (x^(n+1))/(n+1) + C, (n ≠ -1) ∫e^xdx = e^x + C ∫a^xdx = (a^x)/(ln(a)) + C, (a > 0, a ≠ 1) The integral of a power of the natural logarithm: ∫ln(x)dx = xln(x) - x + C The integral of a rational function: ∫(1/x)dx = ln|x| + C