How do you evaluate the integral of  .

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Answer


Sadie

Solution:

Concept: The integration of a function   is the reciprocal of derivative of the function. It is denoted by  .

The anti-derivative of some functions are given in the table below:

 

For two functions in multiplication as  , the integration value in such case is calculated by ILATE method, the formula of which is,

Here, first function u is chosen by ILATE preference where I stands for inverse function, L for logarithmic function, A for algebraic function, T for trigonometric function and E for exponential function.

 

Calculation:

Step:1 The function which is to be integrated is  . Let   such that

Then the integration changes to  .

Step:2 In this, first function is chosen as   then  and  .

Now, substitute the value of u and v in the integration formula as shown below and simplify.

Step:3 Further simplify the above integration expression as follows:

Step:4 Back substitute the value of t as   in the above result as follows:

Hence, the integration of  is  .

 

Answer: The value of integration  is  .

 

 

 

Similar Problems:

 

Problem 1:What is anti-derivative of  ?

Solution:

Step:1 From the Table 1, it is observed the anti-derivative of  is  .

Since anti-derivative is the reciprocal of derivative, the derivative of   must be  .

Verify the result by taking derivative of   as shown below:

Let the function be y such that  .

Step:2 Now take the derivative of the above equation.

Step:3 Simplify further to solve for the derivate as shown below:

Hence, the result obtained from the table is verified and anti-derivative of  is  .

 

Answer: The anti-derivative of  is  .

 

 

 

Problem 2: What is anti-derivative of  ?

Solution:

Step:1  From the Table 1, it is observed the anti-derivative of  is  .

Since anti-derivative is the reciprocal of derivative, the derivative of   must be  .

Verify the result by taking derivative of   as shown below:

Let the function be y such that  .

Step:2 Now take the derivative of the above equation.

Step:3 Simplify further to solve for the derivate as shown below:

 

Hence, the result obtained from the table is verified and anti-derivative of  is  .

 

Answer: The anti-derivative of  is  .

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