Methods Of Integration

To solve the integral , we can use the trigonometric identity: .

This allows us to rewrite the integral as:

Breaking it down, we have:

Calculating the first part:

Calculating the second part:

The integral of is . Therefore:

Putting it all together:

Thus, the integral .

The given integral is:


To solve this integral, we need to recognize that the expression suggests a trigonometric substitution. Let's set , then .

This substitution transforms the integral into:



After simplification, it becomes:



Which simplifies to:



Use the identity :



Splitting the integral, we have:



This evaluates to:



Back substitute , thus . We obtain:



The term involving simplifies to 0 because it fits within the derivative of an antiderivative form.

Therefore, the original integral simplifies beautifully to the abstract form:



Thus, the solution to the integral is:

The problem is to evaluate the integral: To solve this, we'll use substitution and reverse chain rule. Let . Then, or .
Substituting this into the integral: Simplify the expression: The antiderivative of is . Therefore: Substitute back to get: For simplicity, this can be expressed with a positive term: However, checking the options and simplifying further by distributing constants, the appropriate form matching the option is: Hence, the correct answer is: