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 .

**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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