# What is the integration of  ?

Yesenia

Solution:

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

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

Calculation:

Step:1 Let the value   equal to t then the value of   in terms of   is calculated as,

Step:2 Substitute t for  and   for dx in the integral value as shown below:

Step:3 From the Table 1, the integration expression is solved as,

Step:4 Back substitute the value t as   and simplify the above expression.

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  .

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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