`int_(0)^(pi//4) tan^(6)x sec^(2)x dx=`

Options

A. `1/7`
B. `2/7`
C. `1`
D. `3/7`

Answer

To solve the integral \( I = \int_{0}^{\frac{\pi}{4}} \tan^6 x \sec^2 x \, dx \), we will use a substitution method.

### Step-by-Step Solution:

1. **Substitution**: Let \( t = \tan x \). Then, the derivative \( dt = \sec^2 x \, dx \). This means \( dx = \frac{dt}{\sec^2 x} \).

2. **Change of Limits**: When \( x = 0 \), \( t = \tan(0) = 0 \). When \( x = \frac{\pi}{4} \), \( t = \tan\left(\frac{\pi}{4}\right) = 1 \). Therefore, the limits of integration change from \( x = 0 \) to \( x = \frac{\pi}{4} \) into \( t = 0 \) to \( t = 1 \).

3. **Rewrite the Integral**: The integral now becomes:
\[ I = \int_{0}^{1} t^6 \, dt \]

4. **Integrate**: Now we can integrate \( t^6 \):
\[ I = \int_{0}^{1} t^6 \, dt = \left[ \frac{t^{6+1}}{6+1} \right]_{0}^{1} = \left[ \frac{t^7}{7} \right]_{0}^{1} \]

5. **Evaluate the Limits**: Now we evaluate at the limits:
\[ I = \frac{1^7}{7} - \frac{0^7}{7} = \frac{1}{7} - 0 = \frac{1}{7} \]

### Final Answer:
Thus, the value of the integral is:
\[ I = \frac{1}{7} \]