Suppose `int(1-7cos^2x)/(sin^7xcos^2x)dx=(g(x))/(sin^7x)+c` where C is arbitrary constant of integration.then find value of `g'(0)+g''(pi/4)`

Answer

To solve the integral

\[ \int \frac{1 - 7 \cos^2 x}{\sin^7 x \cos^2 x} \, dx \]

we can break it down into two separate integrals:

\[ I = \int \frac{1}{\sin^7 x \cos^2 x} \, dx - 7 \int \frac{\cos^2 x}{\sin^7 x \cos^2 x} \, dx \]

This simplifies to:

\[ I = \int \frac{1}{\sin^7 x \cos^2 x} \, dx - 7 \int \frac{1}{\sin^7 x} \, dx \]

Now, let's denote the first integral as \( I_1 \) and the second as \( I_2 \):

1. **Calculate \( I_1 \)**:

We can use integration by parts for \( I_1 \):

Let \( u = \frac{1}{\sin^7 x} \) and \( dv = \sec^2 x \, dx \).

Then, we have:

\[ du = -\frac{7 \cos x}{\sin^8 x} \, dx, \quad v = \tan x \]

Applying integration by parts:

\[ I_1 = u v - \int v \, du \]

This gives:

\[ I_1 = \frac{\tan x}{\sin^7 x} - \int \tan x \left(-\frac{7 \cos x}{\sin^8 x}\right) \, dx \]

Simplifying, we have:

\[ I_1 = \frac{\tan x}{\sin^7 x} + 7 \int \frac{\tan x \cos x}{\sin^8 x} \, dx \]

The integral \( \int \frac{\tan x \cos x}{\sin^8 x} \, dx \) can be simplified further, but for our purposes, we will focus on the overall structure.

2. **Calculate \( I_2 \)**:

The integral \( I_2 \) is:

\[ I_2 = \int \frac{1}{\sin^7 x} \, dx \]

This integral can be solved using known results or further techniques, but we will focus on the overall result.

Combining both integrals, we find:

\[ I = \frac{\tan x}{\sin^7 x} + C \]

where \( C \) is the constant of integration.

From the question, we have:

\[ \frac{g(x)}{\sin^7 x} + C \]

Comparing both sides, we find:

\[ g(x) = \tan x \]

Now, we need to find \( g'(0) + g''\left(\frac{\pi}{4}\right) \).

3. **Calculate \( g'(x) \)**:

\[ g'(x) = \sec^2 x \]

Evaluating at \( x = 0 \):

\[ g'(0) = \sec^2(0) = 1 \]

4. **Calculate \( g''(x) \)**:

\[ g''(x) = \frac{d}{dx}(\sec^2 x) = 2 \sec^2 x \tan x \]

Evaluating at \( x = \frac{\pi}{4} \):

\[ g''\left(\frac{\pi}{4}\right) = 2 \sec^2\left(\frac{\pi}{4}\right) \tan\left(\frac{\pi}{4}\right) = 2 \cdot 2 \cdot 1 = 4 \]

Finally, we combine the results:

\[ g'(0) + g''\left(\frac{\pi}{4}\right) = 1 + 4 = 5 \]

Thus, the final answer is:

\[ \boxed{5} \]