Integration By Parts

To determine the value of the integral , we can use trigonometric identities and integration techniques.

Start by expressing in terms of :

The integral becomes:

This integral can be split into two separate integrals:

The integral of is , and the integral of is . Therefore, the expression becomes:

where is the constant of integration.

Using the trigonometric identity , rewrite the result:

To further simplify, observe that since:

, it can be related to the identity upon integration, leading to the final result:

This matches with the given options, which confirms the answer: log

To evaluate the integral ∫cos²xdx, we can use the trigonometric identity for cos²x, which is:

cos²x = (1 + cos2x) / 2

Substituting this identity into the integral, we have:

∫cos²xdx = ∫(1 + cos2x)/2 dx

We can split the integral into two separate parts:

= (1/2)∫dx + (1/2)∫cos2xdx

The first integral is straightforward:

(1/2)∫dx = x/2

The second integral requires substituting 2x with a new variable, u. Let:

u = 2x, then du = 2dx, or dx = du/2

Now substitute into the integral:

(1/2)∫cos2xdx = (1/2)∫cosudu/2 = (1/4)∫cosudu

The integral of cosu is sinu, so we have:

(1/4)∫cosudu = (1/4)sinu + C

Re-substitute back for u = 2x:

(1/4)sin2x + C

Combine the results from both integrals:

∫cos²xdx = x/2 + sin2x/4 + C

Therefore, the evaluated integral is:

-x/2 + sin2x/4 + c