Methods Of Integration

Derivative of the given function:
We are asked to find the derivative of the function:

Step 1: Apply the chain rule:
The derivative of is: The derivative of is: The derivative of the constant is 0.

Step 2: Combine the results:
Now, combining all the terms, we get: Simplifying the expression: Using the identity , we get:

Final Answer:
Therefore, the derivative is:

To solve the integral , we will use partial fraction decomposition.
First, decompose the integrand into partial fractions:

Multiplying both sides by the common denominator gives:

Expanding and simplifying the right side:


Combine like terms:

Now, equate the coefficients from both sides:
1. For :
2. For :
3. For constant term:
Solving these equations:
1. implies
2. Substituting into :




3. Substituting and into :



So:



Thus, the partial fractions decomposition is:

Now integrate term-by-term:

The integrals are:
1.
2.
3.
Combining these results, we get:

Simplify using logarithm properties:

Therefore, the final answer is:

1. Simplify the Integrand:
We are given the integral:

2. Factor out from under the square root:
Start by factoring out from under the square root: Simplifying:

3. Divide numerator and denominator by :
Next, divide both the numerator and the denominator by :

4. Make a Substitution:
Let , so that: Notice that:

Now, substituting into the integral:

5. Use Substitution :
Let , and differentiate to get: Now, rewrite the integrand as: Substituting back into the integral:

6. Final Substitution and Simplification:
At this stage, simplify the expression further with proper substitutions: Leading to a more simplified integral that can be solved via standard integration techniques.

Final Answer:
The simplified integral is:

Step 1: Simplify the Integrand using Trigonometric Identities
We are given the integral: Recall that: Substituting this identity into the integral, we get: Simplifying the expression inside the integrand:

Step 2: Recognize the Derivative Relationship
We know that the derivative of with respect to is , i.e.: Let , so that . Therefore, .

Step 3: Rewrite the Integral in Terms of and
Substituting and into the integral, we get: This can be written as:

Step 4: Apply the Integration Rule for
We apply the standard rule for integrating expressions of the form , which states: Since our integral has a negative sign:

Step 5: Substitute to Obtain the Final Result
Now, substitute back into the expression:

Therefore, the integral is:

To integrate , we will integrate each term separately.
1. Integrate with respect to :

2. Integrate with respect to :

3. Integrate with respect to :

Now, combine all these results to find the definite integral:

where is the constant of integration.
So, the evaluated integral is .

To solve the integral we can use the following steps:

1. Rewrite the denominator:
We can rewrite as: Since and , we can write: Using the identity , we get:

2. Substitute into the integral:
Now we substitute this expression into the original integral:

3. Simplify and integrate:
Simplifying the integral: We know that: Therefore, the integral becomes:

Thus, the solution is:

Concept: When the degree of the numerator is equal to or greater than the denominator, we simplify the integrand using algebraic division or decomposition. A useful trick is to rewrite the numerator so that the denominator appears in the expression. can be rewritten as which simplifies the integration process.
Step 1: {Rewrite the numerator.} Thus the integral becomes
Step 2: {Integrate term by term.} Therefore,
Step 3: {Write the final result.}