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Integration: Integration By Parts and Solved Examples

Integration: Integration By Parts and Solved Examples

We all are familiar with the product rule of differentiation. But does there exist some kind of product rule in integration also? Let’s try to find out.

Integration by parts is a technique used to find the integration of the product of two functions. It changes the integration of the product of two functions into integrals for which a solution can be easily computed.

Integration by Parts

Integration by parts is a very important rule for the integration of the product of two functions.

If u and v be two given functions of x, then

image

In words, the above rule can be stated as:

Integral of the product of two functions = (First function) × (integral of the second function) - integral of (dervative of first function × integral of the second function)

Now, the question arises that in the product of two functions which should be considered as first and second. Should they be taken in the order they are given in the question? To answer this question, we have to understand the following points for choosing the first and the second function.

Important points for choosing the first and second functions:

  • If in the product of two functions, one of the functions is not directly integrable (like,
    tan-1x, log|x|), then we take it as the first function and the other function as the second function. For example, In x sin-1x dx, sin-1x is taken as the first function and x is taken as the second function.
  • If there is only one function then unity is taken as the second function, for example, inimage, 1 is taken as the second function.
  • If both functions are integrable, then we can simply use the ILATE rule for such situations.

ILATE Rule

If the integrand contains two different types of functions both of which are directly integrable then the preference order for the first function is according to the ILATE rule. 

Inverse Trigonometric

Logarithmic

Algebraic 

Trigonometric 

Exponential

In the above-stated rule, the function on the top of the order is generally chosen as the first function. 

Related concept video

                    

Example: Evaluate  f  x.sin3x dx

Solution: 

In the given integral, both the functions x(algebraic)  and 3x(trigonometric) are easily integrable, so applying the ILATE rule, x(algebraic) will be the first function as it is above in the priority order as compared to 3x(trigonometric). 

image

Practice Problems of Integration By Parts

Example: Evaluate image

Solution

There is only one function in the integral, so we will consider the other function as unity and then apply the ILATE rule. So, the logarithmic function will be the first function.

image

Example: Evaluate image

Solution

Substitute √x=sin t x=sin2 t so that dx=2 sin t cos t dt=sin 2t dt

image

Example: For x2n+1, n ∈ N, evaluate the integral

image

Solution:

image

Putting x- 1 = ⊖ => 2dx = d⊖, 

image

image

Example: Evaluate image

Solution

image

FAQs of Integration By Parts

Question.1 Which rule of differentiation is used to derive integration by parts formula?

Answer: The product rule of differentiation is used to derive the by parts formula.

Question.2 What are different methods of integration other than integration by parts?

Answer: Other methods are - integration by substitution and integration by partial fractions.

Question.3 How to decide when to use integration by parts?

Answer: Integration by parts is used when the integral contains a product of two or more functions.

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