Integration By Parts

Step 1: Simplify the trigonometric expression. We use the identity: Step 2: Substitute into the integral. Substituting this into the integral: Step 3: Use substitution. Let , so and . The integral becomes: Step 4: Solve the integral. This is a standard integral, and we get: Step 5: Substitute back for . Substitute into the result:

Step 1: Substitution. Let . Differentiating both sides: Thus: Step 2: Substituting into the integral. The integral becomes: Step 3: Solving the integral. The integral of is , so we have: Step 4: Substitute back the value of . Substituting into the result: Since the constant factor of 2 can be simplified, the final result is:

To evaluate the integral , we need to consider the absolute value function, which affects how the integral is computed over different intervals. Step 1: Determine where the expression inside the absolute value equals zero. This point divides the interval into two sub-intervals: Step 2: Rewrite the integral by removing the absolute value, considering the sign of in each sub-interval. 1. For : 2. For : Step 3: Express the original integral as the sum of two integrals over the determined sub-intervals. Step 4: Evaluate each integral separately. 1. First Integral: 2. Second Integral: Step 5: Sum the results of the two integrals to find the total value. Final Answer:

To solve the integral , we follow these steps: Step 1: Rewrite the Integral
We start with the integral: To simplify this, we can multiply and divide by : Step 2: Simplify the Denominator
The denominator can be simplified using the identity : Step 3: Split the Integral
We can split the integral into two parts: Let’s denote these two integrals as and : Step 4: Evaluate
The integral is known to be: Step 5: Evaluate
For , we use the substitution , hence . The limits change from to , which corresponds to to : This evaluates to: Step 6: Combine Results
Now we can combine the results of and : Thus, the value of the integral is:

We are given the expression , and we need to simplify it step by step. Step 1: Express the terms using basic trigonometric identities. We know the following basic trigonometric identities: Therefore, Substituting these into the original expression gives: Step 2: Find an approximation for and . Using known values or a calculator, we approximate: Substitute these values into the expression: Step 3: Simplify the expression. Now, calculate the individual terms: Thus, the expression becomes: Step 4: Conclusion. The value of is approximately 4.

Concept: Use the substitution method when the integrand contains a function and its derivative. Recall the derivative: For a composite function:
Step 1: {Choose substitution.} Let Then
Step 2: {Rewrite the given integral.} Using substitution: Thus the integral becomes
Step 3: {Integrate.}
Step 4: {Substitute back .} Therefore, Hence, the correct option is (C).