Tabular Integration (The tabular method) Examples |Where to use tabular integration by parts method|

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What is Tabular Integration method and when to use?

Tabular integration is a short method for integration to solve the integral problem quickly instead of using the lengthy and tedious process of integration by parts traditional method.

The advantage of the tabular integration method is that it can save huge time in solving the problem and gives the solution fairly accurate than the integration by parts method. This tabular method is fairly quick to memorize and easy to use.

This method is also called the DI method of Integration by parts.

When can I use the Tabular integration method?

The tabular integration method can be applied to any function which is the product of two expressions, where one of the expressions can be differentiated until it gets zero, and another expression can be integrated simultaneously multiple times.

Suppose a function, f(x) = m(x).n(x), from given two expression, one of the expression, let’s take it m(x) should be differentiate multiple times until it reach to zero, and the another expression n(x) should be integrated simultaneously multiple times.

Note: Tabular method can also be used where neither of the expression differentiation goes to zero. Example – (∫e2x.sin3x dx) or (∫sin3x.cos4x dx), etc.

Take a look at where we can apply the tabular integration method.

  • When Integrand is the product of Polynomial times and something that can be repeatedly integrated.  (x10.cos x dx)
  • Integrand multiple of power function and an exponential function. (∫x².ex dx)
  • Integrand multiple of an exponential and trigonometry function. (∫ex.sin x dx)
  • Integrand multiple of power function and a trigonometry function. (∫x³.tan x dx)

How to use or apply the Tabular integration method and its formulas?

how to use or apply tabular integration and its formulas

 

Let the understand by taking the example of the integral function

                    \int _ { } ^ { } t ^ { 2 } \sin t dt

Step 1 = First identify the polynomial function or the function which is differential. So, t² is a polynomial function and denote it as F(x)
F(x) = t²
Step 2 = Now take another product left in function which is integrable and denotes it as F(y) 
F(y) = sint
Step 3 = Create a table and differentiate F(x) until it reaches 0 and simultaneously integrates F(y).
Step 4 = Now change the sign of every second term of the F(x) function to negative Or Add alternating “+” and “-” signs, starting with “+” on the derivatives side of the table.
 

Tabular Integration Method

Now we have, F(x) = t2 and F(y) = sint

F(x) DifferentiateF(y) Integration
sin t
2t-cos t
2-sin t
0cos t

Now we will add alternating “+” and “-” signs, starting with “+” on the derivatives side of the table.

F(x) DifferentiateF(y) Integration
+ t²sin t
– 2t-cos t
+ 2-sin t
 0cos t

Now Most important Steps in the tabular integration method to find the integral of a given function

  1. Multiply F(x) with the first integration of F(y)
  2. Multiply the first derivative of F(x) with the second integration of F(y)………and so on.
  3. Add all of them

example of tabular integration table

So, \int _ { } ^ { } t ^ { 2 } \sin t dt = t²(-cost) + (-2t)(-sint) + 2cost + C

     = -t²cost + 2tsint + 2cost + C   Answer

Tabular Integration Method for solving Definite Integral with limits problems

The definite integral is those integrals that have upper and lower limits or bounds or boundaries. In short, definite integral has start and end values.

Now, How to solve definite integral problems by the Tabular Integration rather than the traditional method and save our time.

Example 1:   

Tabular integration of t3.sint

Step 1 = First identify the polynomial function or the function which is differential. So, t³ is a polynomial function and denote it as F(x)
F(x) = t³
Step 2 = Now take another product left in function which is integrable and denotes it as F(y) 
F(y) = sin t
Step 3 = Create a table and differentiate F(x) until you reach 0 and simultaneously integrate F(y).
Step 4 = Now change the sign of every second term of the F(x) function to negative Or add alternating “+” and “-” signs, starting with “+”:
Now we have, F(x) = t3 and F(y) = sint
F(x) Derivative FunctionF(y) Integration Function
(+)   t³sin t
(-)   3t²-cos t
(+)   6t-sin t
(-)   6cos t
0sin t

Follow this procedure:

  • Multiply F(x) with the first integration of F(y)
  • Multiply the first derivative of F(x) with the second integration of F(y)………and so on.
  • Add all of them

solving definite integral problems with tabular method

\int _ { 0 } ^ { \pi } t ^ { 3 } \sin t dt   = t³( -cost ) \Big| _ 0 ^ \pi – 3t² ( -sint ) \Big| _ 0 ^ \pi + 6tcost \Big| _ 0 ^ \pi   – 6sint \Big| _ 0 ^ \pi  

= -t³cost \Big| _ 0 ^ \pi   + 3t²sint \Big| _ 0 ^ \pi   + 6tcost \Big| _ 0 ^ \pi   – 6(sint) \Big| _ 0 ^ \pi  

= [-π³cosπ – 0] + [3π².0 – 0] + [6πcosπ – o] + [-6(0 – 0]

= -π³(-1) + 6π(-1) + 0

= π³ – 6π  Answer

So, Definite integral of the function \int _ { 0 } ^ { \pi } t ^ { 3 } \sin t dt by tabular integration by parts method is π³ – 6π

Example 2:

tabular integration by parts for definite integral tcos2t

Solution: Using same procedure, Make F(x) = t   and   F(y) = Cos2t

Now construct the table and take appropriate action to solve these definite integral problems by the tabular method

F(x) Derivative FunctionF(y) Integration Function
(+)   tCos2t
(-)   1(1/2)Sin2t
(+)   0-(1/4)Cos2t

Follow these steps for further solving definite integral by the tabular method

  1. Multiply F(x) with the first integration of F(y)
  2. Multiply the first derivative of F(x) with the second integration of F(y)………and so on.
  3. Add all of them

tcos2t integration problem from tabular method

\int _ { 0 } ^ { \frac { \pi } { 4 } } t \cos2t dt = t/2Sin2t \Big| _ 0 ^ \frac { \pi } { 4 } + 1/4Cos2t \Big| _ 0 ^ \frac { \pi } { 4 }  

= [ \frac { \pi } { 8 } \sin \frac { \pi } { 2 } – 0 ]   + \frac { 1 } { 4 } [ \cos \frac { \pi } { 2 } – \cos0 ]  

= \frac { \pi } { 8 } + \frac { 1 } { 4 } ( 0 – 1 )

= \frac { \pi – 2 } { 8 }    Answer

Tabular Integration Examples with their Solutions

Let’s take a few more examples to understand the tabular integration method completely.

Example 1:

Solving problems based on exponential and trigonometric functions using Tabular Integration.

repeated integration by parts problem solving from tabular method

Solution: Now you are thinking that there is no function in the above example that comes up 0 after derivating it endlessly. Then how can we use the tabular integration method here?

Don’t worry just follow the good approach.

Assume, F(x) = sin3t  and  F(y) = e2t

Construct tabular method table for integrating this problem

F(x) Derivative FunctionF(y) Integration Function
Sin(3t)    (+)e2t
3cos(3t)   (-)1/2(e2t)
-9sin(3t)   (+)1/4(e2t)

Now Follow these steps:

  1. Multiply F(x) with the first integration of F(y)
  2. Multiply the first derivative of F(x) with the second integration of F(y).
  3. Multiply the second derivative of F(x) with the second integration of F(y).
  4. Add all of them

e2t.sin3t by tabular method

So, \int _ { } ^ { } e ^ { 2t } \sin3t dt = \frac { 1 } { 2 } e2t sin3t – \frac { 3 } { 4 }  e2t cos3t – \frac { 9 } { 4 } \int _ { } ^ { } e ^ { 2t } \sin3t dt

\frac { 13 } { 4 } \int _ { } ^ { } e ^ { 2t } \sin3t  dt = \frac { 1 } { 2 } e2t sin3t – \frac { 3 } { 4 } e2t cos3t + C

\int _ { } ^ { } e ^ { 2t } \sin3t dt = \frac { 1 } { 13 }  e2t  [ 2sin3t – 3cos3t ] + C   Answer

Example 2: 

practice problem tabular integration by parts method

Solution: let’s give a shot at this integral problem by using the tabular integration method.

Look at the problem and choose a function that will give 0 value after derivating multiple times.

F(x) = t3 – 4t2 + 5t + 32  and   F(y) = sint

Construct the table to solve this problem by Tabular integration method

F(x) Derivative functionF(y) Integration function
(+)     t3 -4t2 + 5t + 32sint
(-)       3t2 – 8t + 5-cost
(+)       6t – 8-sint
(-)           6cost
                0sint

Use same steps

  1. Multiply F(x) with the first integration of F(y)
  2. Multiply the first derivative of F(x) with the second integration of F(y)………and so on.
  3. Add all of them

practice problem DI method

So, we have  \int _ { } ^ { } (t ^ { 3 } – 4t ^ { 2 } + 5t + 32)(sint) dt   = -(t3 – 4t2 + 5t + 32)(cost) + (3t2 – 8t + 5)(sint) + (6t – 8)(cost) – 6sint + C

Simplify form = (3t2 8t 1)sin(t) + (t3 + 4t2 + t40)cos(t) + C     Answer

Example 3:

Solving problems based on power and exponential function using Tabular integration.

tabular integration of exponential problem

Solution: F(x) = t   and    F(y) = e-t

Construct the table to solve this integral problem by tabular integration method

F(x) Derivative FunctionF(y) Integration Function
(+) t5  e-t
(-) 5t-e-t
(+) 20t³e-t
(-) 60t²-e-t
(+) 120te-t
(-) 120-e-t
 (+) 0e-t

Use same steps

  1. Multiply F(x) with the first integration of F(y)
  2. Multiply the first derivative of F(x) with the second integration of F(y)………and so on.
  3. Add all of them

Solving problems based on power and exponential function using Tabular integration

\int _ { } ^ { } t ^ { 5 } e ^ { – t } dt = t5(-e-t) – 5t4(e-t) + 20t3(-e-t) – 60t2(e-t) + 120t(-e-t) – 120(e-t) + C

\int _ { } ^ { } t ^ { 5 } e ^ { – t } dt = -t5(e-t) – 5t4(e-t) – 20t3(e-t) – 60t2(e-t) – 120t(e-t) – 120(e-t) + C  

= (t5 + 5t4 + 20t3 + 60t2 + 120t + 120)et + C    Answer

Example 4:

Solving problem based on two trigonometric function using Tabular integration method

integration by parts tabular method

Solution: We know, in the above problem, both function sin3t and cos4t will not get zero regardless of how much time you can differentiate them.

So, take, any of the functions as F(x) and another one as F(y) and apply the Tabular integration method.

So, I assume,  F(x) = Sin(3t)  and  F(y) = Cos(4t)                [∴You can choose whatever you want]

Now construct the table for solving this by tabular method

F(x) Derivative FunctionF(y) Integration Function
(+)   Sin(3t)Cos(4t)
(-)   3Cos(3t) \frac { 1 } { 4 } Sin(4t)
(+)  -9Sin(3t) – \frac { 1 } { 16 }  Cos(4t)

Use these steps solve this integral problem

  1. Multiply F(x) with the first integration of F(y)
  2. Multiply the first derivative of F(x) with the second integration of F(y).
  3. Multiply the second derivative of F(x) with the second integration of F(y).
  4. Add all of them

tabular integration by parts for repeatedly functions

\int _ { } ^ { } \sin3t \cos4t dt   = \frac { 1 } { 4 } Sin(3t)Sin(4t) + \frac { 3 } { 16 } Cos(3t)Cos(4t) + \frac { 9 } { 16 } \int _ { } ^ { } \sin3t \cos4t dt  

\frac { 7 } { 16 } \int _ { } ^ { } \sin3t \cos4t dt = \frac { 1 } { 4 } Sin(3t)Sin(4t) + \frac { 3 } { 16 } Cos(3t)Cos(4t) + C

\int _ { } ^ { } \sin3t \cos4tdt = \frac { 16 } { 7 } [ \frac { 1 } { 4 }  Sin(3t)Sin(4t) + \frac { 3 } { 16 } Cos(3t)Cos(4t) ] + C   

= [ \frac { 4 } { 7 }  Sin(3t)Sin(4t) + \frac { 3 } { 7 } Cos(3t)Cos(4t) ] + C   Answer

Example 5:

Based on when you have only one function in the problem

tabular integration of Higher degree polynomial problems

Solution: We can solve this problem by assuming as other function 1 and then apply tabular integration method.

So, let F(x) = 80x³    and    F(y) = 1

Construct the table to solve this problem by tabular integration method

F(x) Derivative FunctionF(y) Integration Function
(+) 80x³1
(-)   240x²x
(+)   480xx²/2
(-)   480x³/6
(+)   0x4/24

Use these steps to further solve this integral problem.

  1. Multiply F(x) with the first integration of F(y)
  2. Multiply the first derivative of F(x) with the second integration of F(y)………and so on.
  3. Add all of them

examples of tabular integration method

So, \int _ { } ^ { } 80x ^ { 3 } dx = 80x ^ { 4 } –  240 \frac { x ^ { 4 } } { 2 } + 480 \frac { x ^ { 4 } } { 6 } – 480 \frac { x ^ { 4 } } { 24 }

= 80x ^ { 4 } – 120x ^ { 4 } + 80x ^ { 4 } – 20x ^ { 4 }

= 20x ^ { 4 } + C Answer

Example 6:

tabular integration of x3e-x dx

Solution: Let F(x) = x³    and    F(y) = e-x

Construct the table to solve this problem by tabular integration method

F(x) Derivative FunctionF(y) Integration Function
(+) x³e-x
(-)   3x²– e-x
(+)   6xe-x
(-)   6– e-x
(+)   0e-x

Use these steps to further solve this integral problem.

  1. Multiply F(x) with the first integration of F(y)
  2. Multiply the first derivative of F(x) with the second integration of F(y)………and so on.
  3. Add all of them

problem on tabular method

\int _ { } ^ { } x ^ { 3 } e ^ { – x } dt = +x3(-e-x) – 3x2(e-x) + 6x(-e-x) – 6(e-x) + C

= – x3.e-x – 3x2.e-x – 6x.e-x – 6.e-x + C  Answer

Example 7:

tabular practice problem

Solution: Let F(x) = x   and    F(y) = (x – 2)3/2

Construct the table to solve this problem by tabular integration method

F(x) Derivative FunctionF(y) Integration Function
(+) x2(x – 2)3/2
(-)   2x2/5(x – 2)5/2
(+)   24/35(x – 2)7/2
(-)   08/315(x – 2)9/2

Use these steps to further solve this integral problem.

  1. Multiply F(x) with the first integration of F(y)
  2. Multiply the first derivative of F(x) with the second integration of F(y)………and so on.
  3. Add all of them

tabular integration practice problem

\int_{}^{} x^{2}(x – 2)^{\frac{3}{2}}dx = \frac{2}{5}x^{2}(x-2)^{\frac{5}{2}} – \frac{8}{35}x(x-2)^{\frac{7}{2}}+ \frac{16}{315}(x-2)^{\frac{9}{2}} + C   Answer

Example 8:

tabular integration of cos2x.6x3

Solution: Let F(x) = 6x   and    F(y) = cos(2x)

Construct the table to solve this problem by tabular integration

F(x) Derivative FunctionF(y) Integration Function
(+) 6x3cos(2x)
(-)   18x2sin(2x)/2
(+)   36x-cos(2x)/4
(-)   36-sin(2x)/8
0cos(2x)/16

Use these steps to further solve this integral problem.

  1. Multiply F(x) with the first integration of F(y)
  2. Multiply the first derivative of F(x) with the second integration of F(y)………and so on.
  3. Add all of them

method of integration

\int_{}^{} 6x^{3}.cos(2x)dx = 3x3sin2x + (9/2)x2cos2x – (9/2)xsin2x – (9/4)cos2x + C   Answer

Example 9:

tabular integration of 4x2.e-2x

Solution: Let F(x) = 4x   and    F(y) = e-2x

Construct the table to solve this problem by tabular integration 

F(x) Derivative FunctionF(y) Integration Function
(+) 4x2e-2x
(-)   8x-e-2x/2
(+)   8e-2x/4
(-)   0-e-2x/8

Use these steps to further solve this integral problem.

  1. Multiply F(x) with the first integration of F(y)
  2. Multiply the first derivative of F(x) with the second integration of F(y)………and so on.
  3. Add all of them

integration by parts tabular method

\int_{}^{} 4x^{2}.e^{-2x}dx = -2x2e-2x – 2xe-2x – e-2x + C   Answer

FAQ

What is the Tabular Integration?

The tabular Integration is also called the DI method to solve integration problems quickly by forming three columns, the first one for “Alternative sign”, the second column for “Derivative function” and the third column for “Integration function”.

Integration by parts vs Tabular integration Method?

Integration by parts is the traditional method that is used to finds the integral of a product of functions and Tabular integration is a short technique to solve integral problems quickly by letting one of the functions can be differentiated multiple times and the other function can be integrated multiple times with ease.

When do you can use the tabular integration?

Tabular integration can be used where one function is differentiated until it gets zero, and another function can be integrated simultaneously multiple times. Examples – ∫x6.sinx dx, ∫x3.e4x dx, etc.

Tabular integration can also be used where none of the functions gets zero when differentiated multiple times. Examples – ∫cosx.sinx dx, ∫ex.sin5x dx, etc.

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