What is `inttan^(2)xsec^(4)xdx` equal to ?

Options

A. A) `(sec^(5)x)/(5)+(sec^(3)x)/(3)+c`
B. B) `(tan^(5)x)/(5)+(tan^(3)x)/(3)+c`
C. C) `(tan^(5)x)/(5)+(sec^(3)x)/(3)+c`
D. D) `(sec^(5)x)/(5)+(tan^(3)x)/(3)+c`

Answer

To solve the integral \( \int \tan^2 x \sec^4 x \, dx \), we can follow these steps:

### Step 1: Rewrite the integral
We start with the integral:
\[ I = \int \tan^2 x \sec^4 x \, dx \]
We can rewrite \( \sec^4 x \) as \( \sec^2 x \cdot \sec^2 x \):
\[ I = \int \tan^2 x \sec^2 x \cdot \sec^2 x \, dx \]

### Step 2: Use the identity for secant
Recall the identity:
\[ \sec^2 x = 1 + \tan^2 x \]
Thus, we can substitute \( \sec^2 x \) in the integral:
\[ I = \int \tan^2 x (1 + \tan^2 x) \sec^2 x \, dx \]
This expands to:
\[ I = \int \tan^2 x \sec^2 x \, dx + \int \tan^4 x \sec^2 x \, dx \]

### Step 3: Use substitution
Let \( t = \tan x \). Then, the derivative \( dt = \sec^2 x \, dx \). This implies:
\[ dx = \frac{dt}{\sec^2 x} = \frac{dt}{1 + t^2} \]
Now, substitute \( t \) into the integral:
\[ I = \int t^2 \, dt + \int t^4 \, dt \]

### Step 4: Integrate each term
Now we can integrate each term separately:
1. For \( \int t^2 \, dt \):
\[ \int t^2 \, dt = \frac{t^3}{3} \]
2. For \( \int t^4 \, dt \):
\[ \int t^4 \, dt = \frac{t^5}{5} \]

### Step 5: Combine the results
Combining the results, we have:
\[ I = \frac{t^3}{3} + \frac{t^5}{5} + C \]

### Step 6: Substitute back
Now, substitute back \( t = \tan x \):
\[ I = \frac{\tan^3 x}{3} + \frac{\tan^5 x}{5} + C \]

### Final Result
Thus, the final result for the integral is:
\[ \int \tan^2 x \sec^4 x \, dx = \frac{\tan^3 x}{3} + \frac{\tan^5 x}{5} + C \]

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