Integration of 1/sinx * cosx:

To evaluate the integral of 1/sinx * cosx, we can utilize trigonometric identities and substitution techniques. Let's break down the process step by step:

Step 1: Simplify the expression:
To simplify the given expression, we can use the identity: sin^2(x) + cos^2(x) = 1. Rearranging this equation, we get cos^2(x) = 1 - sin^2(x). Dividing both sides by cos^2(x), we obtain 1/cos^2(x) = 1/(1 - sin^2(x)).

Now, we can rewrite the integral as ∫(1/sinx * 1/cos^2x) dx.

Step 2: Apply a substitution:
Let's substitute sinx with a new variable u. We set u = sinx, then differentiate both sides with respect to x, du/dx = cosx. Rearranging this equation, we get dx = du/cosx.

Substituting dx and sinx with the derived expressions, we can rewrite our integral as ∫(1/u * 1/(1 - u^2)) du/cosx.

Step 3: Simplify the integral:
Now, we can simplify the integral further by canceling out cosx from the numerator and denominator, which leaves us with ∫(1/u * 1/(1 - u^2)) du.

Step 4: Use partial fraction decomposition:
To integrate the expression ∫(1/u * 1/(1 - u^2)) du, we can apply partial fraction decomposition. We express 1/(u * (1 - u^2)) as A/u + (B/(1 - u)) + (C/(1 + u)), where A, B, and C are constants.

Step 5: Determine the values of A, B, and C:
To find the values of A, B, and C, we can equate the numerator on both sides of the equation. After solving the equation, we find A = 1/2, B = 1/2, and C = -1/2.

Step 6: Integrate the partial fractions:
Now, we can integrate each partial fraction separately. The integral of A/u is A ln|u|, the integral of B/(1 - u) is -B ln|1 - u|, and the integral of C/(1 + u) is C ln|1 + u|.

Step 7: Combine the integrals:
Combining the integrals of the partial fractions, we have A ln|u| - B ln|1 - u| + C ln|1 + u|.

Step 8: Substitute back to the original variable:
Finally, we substitute back u with sinx in the expression A ln|u| - B ln|1 - u| + C ln|1 + u|. This gives us the final result of the integral.

Conclusion:
The integral of 1/sinx * cosx is A ln|sinx| - B ln|1 - sinx| + C ln|1 + sinx|, where A