Integration

Step 1: Understanding the integrals and derivatives.
- (i) The integral of is based on the standard integral formula for . - (ii) The integral of is . - (iii) The integral of is . - (iv) The integral of is . - (v) The integral of is , as this is the standard form of the cotangent integral. - (vi) The derivative of is , using the chain rule for the derivative of a sine function.

Step 2: Conclusion.
The correct matches are:

Step 1: Symmetry of the integral.
Observe the symmetry in the integrand. By substituting , we can transform this integral into a simpler form.

Step 2: Simplifying the integral.
Use the symmetry property of definite integrals and trigonometric identities to simplify the integral:

Step 3: Combining both integrals.
By adding the two integrals, we get: Thus, each integral equals half of , which is .

Step 4: Conclusion.
Therefore, the value of the integral is .

Step 1: Break the integral into regions based on the absolute value.
We need to split the integral at because the absolute value function behaves differently for positive and negative values of . Thus, we can write:

Step 2: Calculate each part of the integral.
- First integral: - Second integral:

Step 3: Combine the results.
Now, add the results from both integrals:

Step 4: Conclusion.
Thus, the value of the integral is:

Step 1: Understand the integral.
The given integral is: This is an even function because the powers of and are both odd and even, respectively. Therefore, we can evaluate the integral over and then multiply by 2.

Step 2: Solution.
The exact evaluation would require applying standard methods, possibly substitution or known reduction formulas, but it's clear from symmetry and integral properties that the result is zero (since the function is odd over symmetric limits).

Step 3: Conclusion.
Thus, the value of the integral is:

Step 1: Apply the product rule.
Using the product rule of differentiation in reverse, we get:

Step 2: Conclusion.
Thus, .

Step 1: Using trigonometric identity.
We know that , so the integral becomes:

Step 2: Conclusion.
Thus, .