Integrate

Integrate: ∫ csc(x) cot(x) – sec^2(x) dx

The Correct Answer and Explanation is:

To solve the integral ∫csc⁡(x)cot⁡(x)−sec⁡2(x) dx\int \csc(x) \cot(x) – \sec^2(x) \, dx∫csc(x)cot(x)−sec2(x)dx, we will break it into two simpler integrals.

Step 1: Separate the integral

We can split the integral as follows:∫csc⁡(x)cot⁡(x)−sec⁡2(x) dx=∫csc⁡(x)cot⁡(x) dx−∫sec⁡2(x) dx\int \csc(x) \cot(x) – \sec^2(x) \, dx = \int \csc(x) \cot(x) \, dx – \int \sec^2(x) \, dx∫csc(x)cot(x)−sec2(x)dx=∫csc(x)cot(x)dx−∫sec2(x)dx

Step 2: Solve the first integral ∫csc⁡(x)cot⁡(x) dx\int \csc(x) \cot(x) \, dx∫csc(x)cot(x)dx

Recall that the derivative of csc⁡(x)\csc(x)csc(x) is −csc⁡(x)cot⁡(x)-\csc(x) \cot(x)−csc(x)cot(x), which is useful for recognizing the integral.

The integral of csc⁡(x)cot⁡(x)\csc(x) \cot(x)csc(x)cot(x) is simply −csc⁡(x)-\csc(x)−csc(x), since:∫csc⁡(x)cot⁡(x) dx=−csc⁡(x)\int \csc(x) \cot(x) \, dx = -\csc(x)∫csc(x)cot(x)dx=−csc(x)

Step 3: Solve the second integral ∫sec⁡2(x) dx\int \sec^2(x) \, dx∫sec2(x)dx

The integral of sec⁡2(x)\sec^2(x)sec2(x) is a standard result:∫sec⁡2(x) dx=tan⁡(x)\int \sec^2(x) \, dx = \tan(x)∫sec2(x)dx=tan(x)

Step 4: Combine the results

Now, we can combine the results from both integrals:∫csc⁡(x)cot⁡(x)−sec⁡2(x) dx=−csc⁡(x)−tan⁡(x)+C\int \csc(x) \cot(x) – \sec^2(x) \, dx = -\csc(x) – \tan(x) + C∫csc(x)cot(x)−sec2(x)dx=−csc(x)−tan(x)+C

Where CCC is the constant of integration.

Final Answer:

−csc⁡(x)−tan⁡(x)+C-\csc(x) – \tan(x) + C−csc(x)−tan(x)+C

Explanation:

• The first integral ∫csc⁡(x)cot⁡(x) dx\int \csc(x) \cot(x) \, dx∫csc(x)cot(x)dx was recognized by its similarity to the derivative of csc⁡(x)\csc(x)csc(x).
• The second integral ∫sec⁡2(x) dx\int \sec^2(x) \, dx∫sec2(x)dx was solved using the standard formula for the derivative of tan⁡(x)\tan(x)tan(x).
• By combining both, the result was obtained.