IIT JEE Integration by Parts

Properties of Indefinite Integral for JEE Main and Advanced

Indefinite integral:

An Integral form ∫f(z)dz without upper and lower limits is also called an anti-derivative. The theorem of calculus allows definite integrals to be computed in terms of indefinite integrals. As per this theorem, it is stated that if F is the indefinite integral for a complex function f(z), then ∫abf(z)dz=F(b)-F(a)

This result is the purely algebraic indefinite integral and analytic (or geometric) definite integral. Indefinite integration is implemented in [f, z].

Since the derivative of a constant is zero so any constant may be added to an anti-derivative and it will still correspond to the same integral. Another method we use to the anti-derivative is a non-unique inverse of the derivative. For this, indefinite integrals are often written in the form

∫f(z)dz=F(z)+c

Where, C is a constant of integration (This is called an arbitrary constant).

  • • Indefinite integration is also termed as anti-differentiation and it is the reverse process of differentiation. This is an anti-derivative of function f(x).

  • Let we have a function f and we intend to find a function F such that F’ = f, then in this case F is said to be the indefinite integral or the anti-derivative of the function f.

    Mathematically,
        F(x) = ∫f(x)dx

    Where, f(x) : is called the integrand,
    X: integration variable and ‘c’: constant of integration (i.e. an arbitrary constant).
    i.e. F(x)= ∫f(x)dx iff F(x)= ∫f(x)

  • • Inverse property of indefinite integrals:


  • 1. d/dx ∫ f(x) dx= f(x)

  • 2. ∫ f’(x) dx = f(x) + c

  • If F is an anti-derivative of a continuous function f, then any other anti-derivative G of the function f is of the form G(x) = F(x) + c. So, two anti-derivatives of a function will only differ by a constant.

    If ∫ f(x) dx = g(x) + c, then ∫ f(ax + b) dx = 1/a g(ax + b) + c, where a and b are constants such that a ≠ 0.

  • • Constant is a component in indefinite integration. 

  • .
    Properties of Indefinite Integral:

    In this part we will extract some very important properties of indefinite Integral which will be going to help us to understand this concept with ease.

  • 1) First property says that the process of integration and differentiation are inverse/contrary to each other. Let’s get this thing with the sense of the following results:

  • d/dx ∫f (x) dx = f (x) and
    ∫ f’ (x) dx = f (x) + C , and here C is any arbitrary constant.

    For proving this thing, let’s take one aspect;

    Proof-

    Suppose F could be any kind of anti-derivative of f, i.e.
    d/dx F (x) = f (x)

    So,

    ∫ f (x) dx = F (x) + C

    Thus,

    d/dx ∫ f (x) dx = d/dx (F (x) + C)
           = d/dx F(x) = f (x)

    Correspondingly, we account that
                    f’ (x) = d/dx f (x)

    And as a result ∫ f’ (x) dx = f (x) + C

    Here, C is any arbitrary constant which is called constant of integration.

  • 2) In the second property, it says that 2 integrals which lead to the same family of curves, with the same derivative are equivalent. Let’s understand this with the following results:

  • Proof-

    Suppose g and f be 2 functions such that
              d/dx ∫ f (x) dx = d/dx ∫ g (x) dx

    Or d/dx [ ∫ f (x) dx - ∫ g (x) dx ] = 0

    Therefore,

     ∫ f (x) dx - ∫ g (x) dx = C, here C is any kind of real number (why?)

    Or ∫ f (x) dx = ∫ g (x) dx + C

    Thus, the families of curves { ∫ f (x) dx + C1, C1 ∈ R }

    Also { ∫ g (x) dx + C2, C2 ∈ R } are identical.

    Thus, in that sense, ∫ f (x) dx and ∫ g (x) dx are equivalent.

    NOTE: The equivalence of the families {∫ f (x) dx + C1, C1 ∈ R } in addition to { ∫ g (x) dx + C2 ,C2 R } is conventionally expressed by writing ∫ f (x) dx = ∫ g (x) dx , without any need to mention the parameter.

  • 3) In the third property, it is mentioned that,

  •                     ∫ f (x) + g (x)] dx = ∫ f (x) dx + g (x) dx

    Proof- After drawing from the property (1), we have

                  d/dx [ ∫ [ f (x) + g (x) ] dx ] = f (x) + g (x) ….(I)

    On the contrary, we get that

     d/dx [ ∫ f (x) dx + ∫ g (x) dx ] = d/dx ∫ f (x) dx + d/dx ∫ g (x) dx ….(II)

    Therefore, in the context of property (2), it must follow by (I) and (II) that

              ∫ (f (x) + g (x) ) dx = ∫ f (x) dx + ∫ g (x) dx.

  • 4) In the fourth property, it is stated that for any kind of real number k, ∫ k f (x) dx = k ∫ f (x) dx

  • Proof- After drawing from the property (1), d/dx ∫ k f (x) dx = k f (x).

    In addition,

    d/dx [ k ∫ f (x) dx ] = k d/dx ∫ f (x) dx = k f (x)

    Thus, by using the property (2), we get ∫ k f (x) dx = k ∫ f (x) dx.

  • 5) In the fifth property, it is being said property (3) and property (4) can be induced to a finite number of the real numbers k1, k2,……………., kn and finite number of functions f1, f2,……………, fn giving

  • ∫ [k1 f1 (x) + k2 f2 (x) + …….. + kn fn (x) ] dx
    = k1 ∫ f1 (x) dx + k2 ∫ f2 (x) dx +……..+ kn ∫ fn (x) dx.

    To detect any kind of or any anti-derivative of a given function, we search instinctively for a function of whom derivative is the given function. Integration by the method of inspection is a process where the search is done for the requisite function for finding an anti derivative.

    Integration by Parts

    If u and v are two functions of x, then the integral of the product of these two functions is given below:



    Note: In applying the above equation, it has to be taken in the selection of the first function (u) and the second function (v) depending on which function can be integrated easily.

    We use the following methods for making this choice:

    If both of the functions are directly integrated then the first function is chosen in such a way that the derivative of the function thus obtained under integral sign is easily integrated. Normally we use the preference order for the first function i.e. ILATE RULE (Inverse, Logarithmic, Algebraic, Trigonometric, Exponent) which states that the inverse function should be assumed as the first function while performing the integration. Hence, the functions are to be assumed in the order from left to right depending on the type of functions involved.

    A useful rule of integral by parts is ILATE. There are following steps for u.

  • IInverse trigonometric functions : sin-1(x), cos-1(x), tan-1(x)

  • LLogarithmic functions : ln(x), log(x)

  • AAlgebraic functions : x2, x3

  • TTrigonometric functions : sin(x), cos(x), tan (x)

  • EExponential functions : ex, 3x

  •  
    Example:

    Evaluate ∫ex sin(x) dx

    Solution:

    Let u = sin(x)
    v = ex

    Differentiation of u: sin(x)' = cos(x)

    Integration of v: ∫ex dx = ex

    Now put it together:

    ∫ex sin(x) dx = sin(x) ex -∫cos(x) ex dx

    For the above equation, we can use integration by parts again:

    Choose u and v:

  • • u = cos(x)

  • • v = ex

  • Differentiate u: cos(x)' = -sin(x)

    Integrate v: ∫ex dx = ex

    Now put it together:

    ∫ex sin(x) dx = sin(x) ex - (cos(x) ex −∫−sin(x) ex dx)

    Simplify:

    ∫ex sin(x) dx = ex sin(x) - ex cos(x) −∫ ex sin(x)dx

    Now we have the same integral on both sides. So, bring the right hand one over to the left to get:

    2∫ex sin(x) dx = ex sin(x) − ex cos(x)

    Simplify:

    ∫ex sin(x) dx = ex (sin(x) - cos(x)) / 2 + C

    Definite Integrals

    Definite Integral is a type of integral which has upper and lower limit or you can say it has the values of “start” and “end”. Here, 
    we can deduce that there is an interval “a” and “b” and these intervals are called boundaries or limits.

    We can understand this thing by given formula: ∫ab

    Now we discuss the integration by parts formula for definite integrals.

    ba udv = uv]ba - ∫ba vdu

    At this point we must understand that uv]ba is just the standard integral evaluation notation in the first term.