The indefinite integral `I=int((sin^(2)x-cos^(2)x)^(2019))/((sinx)^(2021)(cosx)^(2021))dx` simplifies to (where c is an integration constant)

Options

A. `((sin^(2)x-cos^(2)x)^(2020))/(2020)+c`
B. `((tanx-cotx)^(2020))/(2020)+c`
C. `((sinx-cosx)^(2020))/(2020)+c`
D. `((tan^(2)x+cot^(2)x)^(2020))/(2020)+c`

Answer

To solve the indefinite integral

\[ I = \int \frac{(\sin^2 x - \cos^2 x)^{2019}}{(\sin x)^{2021} (\cos x)^{2021}} \, dx, \]

we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integral:

\[ I = \int \frac{(\sin^2 x - \cos^2 x)^{2019}}{(\sin x)^{2021} (\cos x)^{2021}} \, dx. \]

### Step 2: Factor the Numerator
Notice that we can factor out \((\sin x)^{2019}\) and \((\cos x)^{2019}\) from the numerator:

\[ I = \int \frac{(\sin^2 x - \cos^2 x)}{(\sin x \cos x)^{2021}} \cdot \frac{(\sin x)^{2019} (\cos x)^{2019}}{(\sin x \cos x)^{2019}} \, dx. \]

This simplifies to:

\[ I = \int \frac{(\sin^2 x - \cos^2 x)}{(\sin x \cos x)^{2021}} \cdot \frac{1}{(\sin^2 x \cos^2 x)} \, dx. \]

### Step 3: Use the Identity
Using the identity \(\sin^2 x + \cos^2 x = 1\), we can rewrite the numerator:

\[ \sin^2 x - \cos^2 x = -(\cos^2 x - \sin^2 x) = -\cos^2 x + \sin^2 x. \]

Thus, we can express the integral as:

\[ I = \int \frac{\sin^2 x - \cos^2 x}{\sin^2 x \cos^2 x} \cdot \frac{1}{(\sin x \cos x)^{2019}} \, dx. \]

### Step 4: Change of Variables
Let \(t = \tan x - \cot x\). Then, we differentiate:

\[ dt = \left(\sec^2 x + \csc^2 x\right) \, dx. \]

This gives us:

\[ dx = \frac{dt}{\sec^2 x + \csc^2 x}. \]

### Step 5: Substitute and Integrate
Substituting back into the integral, we have:

\[ I = \int t^{2019} \cdot \frac{1}{\sec^2 x + \csc^2 x} \, dt. \]

The integral simplifies to:

\[ I = \int t^{2019} \, dt. \]

### Step 6: Solve the Integral
Now we can integrate:

\[ I = \frac{t^{2020}}{2020} + C. \]

### Step 7: Substitute Back
Substituting back the value of \(t\):

\[ I = \frac{(\tan x - \cot x)^{2020}}{2020} + C. \]

### Final Answer
Thus, the indefinite integral simplifies to:

\[ I = \frac{(\tan x - \cot x)^{2020}}{2020} + C. \]