# How do you evaluate the integral of .  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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