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 .

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** **

**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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