`int(sin^(m)x)/(cos^(m+2)x)dx=`

Options

A. `(tan^(m+1))/(m+1)+c`
B. `(tan^(m)x)/(m)+c`
C. `tan^(m)x+c`
D. `(m)/(m+1)log(tanx)+c`

Answer

To solve the integral \( \int \frac{\sin^m x}{\cos^{m+2} x} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integral:
\[ \int \frac{\sin^m x}{\cos^{m+2} x} \, dx = \int \frac{\sin^m x}{\cos^m x \cdot \cos^2 x} \, dx \]

### Step 2: Use Trigonometric Identities
Using the identity \( \tan x = \frac{\sin x}{\cos x} \), we can express the integral as:
\[ \int \tan^m x \cdot \frac{1}{\cos^2 x} \, dx \]
Since \( \frac{1}{\cos^2 x} = \sec^2 x \), we can rewrite the integral as:
\[ \int \tan^m x \cdot \sec^2 x \, dx \]

### Step 3: Substitution
Now we can use the substitution \( t = \tan x \). The derivative of \( \tan x \) is \( \sec^2 x \), which gives us \( dt = \sec^2 x \, dx \). Thus, we can rewrite the integral in terms of \( t \):
\[ \int t^m \, dt \]

### Step 4: Integrate
Now we can integrate using the power rule:
\[ \int t^m \, dt = \frac{t^{m+1}}{m+1} + C \]

### Step 5: Back Substitute
Finally, we substitute back \( t = \tan x \):
\[ \frac{\tan^{m+1} x}{m+1} + C \]

### Final Answer
Thus, the solution to the integral is:
\[ \int \frac{\sin^m x}{\cos^{m+2} x} \, dx = \frac{\tan^{m+1} x}{m+1} + C \]

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