Evaluate the following integrals.
`int tanx sec^(6) x dx `

Answer

To evaluate the integral \( \int \tan x \sec^6 x \, dx \), we can use the method of substitution. Here’s a step-by-step solution:

### Step 1: Choose a substitution
Let us choose:
\[ t = \sec x \]
Then, we know that:
\[ \frac{dt}{dx} = \sec x \tan x \quad \Rightarrow \quad dt = \sec x \tan x \, dx \]

### Step 2: Rewrite the integral
We can express \( \tan x \sec^6 x \, dx \) in terms of \( t \):
\[ \tan x \sec^6 x \, dx = \tan x \sec^6 x \cdot \frac{dt}{\sec x \tan x} = \sec^5 x \, dt \]
Since \( \sec x = t \), we have:
\[ \sec^5 x = t^5 \]
Thus, the integral becomes:
\[ \int \tan x \sec^6 x \, dx = \int t^5 \, dt \]

### Step 3: Integrate
Now we can integrate:
\[ \int t^5 \, dt = \frac{t^6}{6} + C \]

### Step 4: Substitute back
Now we substitute back \( t = \sec x \):
\[ \frac{t^6}{6} + C = \frac{\sec^6 x}{6} + C \]

### Final Answer
Thus, the final answer is:
\[ \int \tan x \sec^6 x \, dx = \frac{\sec^6 x}{6} + C \]

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