Methods Of Integration

We are asked to evaluate the integral:

Step 1: Expand the integrand First, we expand the square of the binomial inside the integral: Thus, the integral becomes: \

Step 2: Simplify using trigonometric identities We can simplify by using the identity : Now, substitute this expression back into the integral: Next, simplify by using the identity again: Thus, the integral becomes:

Step 3: Break the integral into simpler terms Now, we break the integral into separate terms: Each of these integrals can be solved using standard integration techniques.

Step 4: Solve the integrals We now solve each integral separately. 1. For , we can use the reduction formula for powers of sine: 2. For , use the reduction formula for odd powers of sine: 3. Similarly, for : 4. The integral . 5. For , use the identity : 6. Finally, for , use the reduction formula:

Step 5: Add all terms together Now, adding all the terms together: Thus, the final answer is:

We are given two curves: We need to find the area of the region enclosed by these curves.

Step 1: Find the points of intersection To find the points of intersection, we set the two equations equal to each other: Expanding both sides: Simplifying: So, the curves intersect at .

Step 2: Set up the integral for the area To find the area between the curves, we subtract the lower curve from the upper curve. Since both curves are in terms of , we express in terms of . From the equation , we get: From the equation , we get: Thus, the area is given by the integral of the difference of the upper and lower functions over the range of from to . The integral for the area is:

Step 3: Solve the integral We now solve the integrals separately: 1. For , make the substitution . This gives: Evaluating this from to . 2. For , make the substitution , and we get: Evaluating this from to .

Step 4: Calculate the area Finally, we combine the results to find the area, which simplifies to: Thus, the area of the region enclosed by the curves is . \bigskip