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Integration

Real life applications of integration Integration is used for many real-life applications. Many derivations in physics and chemistry are done using integration. Say for example to calculate the half-life of a radioactive substance. It is also used in some parts of biology.

Integration is also used in many business related applications like: finding the cost function from marginal cost function, finding the revenue function from marginal revenue function, etc.
Integration Rules

Anti-differentiation

Integration is an inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and asked to find its original function. Such a process is called integration or anti-differentiation. Therefore, the primary object of differential calculus is: 'given a function, to find its differential coefficient' whereas the primary object of integral calculus is its inverse, i.e., 'given the differential coefficient of a function, to find the function itself'.

Definition: Let f(x) be the given function of 'x'. The family of all anti-derivatives of a function f(x) is called the indefinite integral of f with respect to x and is denoted by f(x) dx.

If φ(x) is any function such that φ'(x) = f(x), then f(x) dx = φ(x) + c, where 'c' is an arbitrary constant, called the constant of integration.

In the notation: f(x) dx, the symbol was introduced by Leibniz and is called the integral sign, the function to be integrated, i.e., f(x) is called the integrand, and 'dx' indicates that 'x' is the variable of integration.

Standard forms: We know the differential coefficients of standard functions. Therefore, below are the list of some elementary standard forms of integrals which are obtained from those known results.

Differentiation formulae Integration formulae
(x) = 1 1 dx = x + c
(kx) = k k dx = kx + c
(xn + 1) = (n + 1)xn
(sin x) = cos x cos x dx = sin x + c
(cos x) = – sin x sin x dx = – cos x + c
(tan x) = sec2 x sec2 x dx = tan x + c
(cot x) = – cosec2 x cosec2 x dx = – cot x + c
(sec x) = sec x tan x sec x tan x dx = sec x + c
(cosec x) = – cosec x cot x cosec x cot x dx = – cosec x + c
(ln x) = dx = ln |x| + c
(ex) = ex ex dx = ex + c
(ax) = (ln a)ax ax dx = + c, a > 1
(Sin– 1 x) = dx = Sin– 1 x + c
(Tan– 1 x) = dx = Tan– 1 x + c
(Sec– 1 x) = dx = Sec– 1 x + c
Integration rules

Constant rule: k dx = kx + c. This rule states that the indefinite integral of a constant function with value 'k' is kx + c.

Constant multiple rule: k.f(x) dx = k f(x) dx. This rule states that the indefinite integral of the constant times a function is the constant times the indefinite integral of the function.

  • Ex: Find the integral of the functions: (i) f(x) = 6x2 and (ii) f(x) = 4x3.

    Sol:

Sum rule: [f(x) + g(x)] dx = f(x) dx + g(x) dx. This rule states that the indefinite integral of the sum of two functions is the sum of the indefinite integrals of two functions.

  • Ex: Find the integral of sum of two functions: f(x) = 3x5 and g(x) = 4x7.

    Sol:

Difference rule: [f(x) – g(x)] dx = f(x) dx – g(x) dx. This rule states that the indefinite integral of the difference of two functions is the difference of the indefinite integrals of two functions.

  • Ex: Find the integral of difference of two functions: f(x) = x3 and g(x) = x4.

    Sol:

Power rule: . This rule states that when integrating a power of 'x', we add one to the exponent and then divide by the new exponent.

Integral of a derivative of a function: f(x) dx = F(x) + c ⇔ F'(x) = f(x). From this, we conclude that the integral of the derivative of a given function is equal to the given function only.

Integration by Substitution

Integrals of certain functions cannot be obtained directly if they are not in one of the standard forms given, but these functions may be reduced to standard forms by proper substitution. This method of evaluating an integral by reducing it to standard form by a proper substitution is known as integration by substitution.

Procedure for making U-substitution
If the function g(x) is a continuously differentiable function, then to evaluate the integrals of the form f(g(x)) dx, we have the following steps:
Step 1: Let u = g(x).
Step 2: Differentiate: du = g'(x) dx.
Step 3: Rewrite the integral in terms of u.
Step 4: Evaluate the integral.
Step 5: Replace 'u' by g(x).
Step 6: Check the result by taking the derivative of the answer.
  • Ex 1: Use integration by substitution to find cos (x2) 2x dx.

    Sol:
    Step 1: Let u = x2. Step 2: Differentiate: du = 2x dx.
    Step 3: Rewrite: cos u du. Step 4: Integrate: sin u + c.
    Step 5: Replace u: sin (x2) + c. Step 6: Differentiate and check:
  • Ex 2: Use integration by substitution to find x2 dx.

    Sol:
    Step 1: Let u = x3. Step 2: Differentiate: du = 3x2 dx ⇒ du = x2 dx.
    Step 3: Rewrite: eu du. Step 4: Integrate: eu + c.
    Step 5: Replace u: + c. Step 6: Differentiate and check:

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