Integrating sin(x) cos(x) and Complex Integrals

When dealing with integrals, one of the most fundamental operations is integrating products of trigonometric functions, such as sin(x) cos(x).

Basic Integration of sin(x) cos(x)

The integral of (sin x cos x) can be solved by recognizing that the derivative of (sin x) is (cos x), and vice versa. Therefore, the antiderivative of (sin x cos x) can be expressed as:

[int sin x cos x , dx -cos x sin x C]

Integration Steps

To break down the integration step-by-step:

Step 1: Recognize that (int sin x cos x , dx)) can be seen as the product of antiderivatives:

[int sin x cos x , dx int sin x , dx cdot int cos x , dx]

Step 2: Compute the individual antiderivatives:

(int sin x , dx -cos x C_1)

(int cos x , dx sin x C_2)

Step 3: Multiply the antiderivatives:

[int sin x cos x , dx (-cos x C_1)(sin x C_2)]

Step 4: Simplify the expression:

[int sin x cos x , dx -cos x sin x C]

Note: The constants (C_1) and (C_2) are combined into a single constant (C).

Challenging Integral: (int frac{sqrt{sin x}}{sqrt{sin x}sqrt{cos x}}, dx)

This particular integral involves a more complex trigonometric expression. Let's break down the steps:

Step 1: Simplify the Expression

The given integral is:

[I int frac{sqrt{sin x}}{sqrt{sin x}sqrt{cos x}} , dx]

By simplifying the expression, we get:

[I int frac{sqrt{sin x}}{sqrt{sin x}sqrt{cos x}} , dx int frac{1}{sqrt{cos x}} , dx]

Step 2: Substitution

Substitute (tan x t^2) and (sec^2 x , dx 2t , dt) to further simplify:

[dx frac{2t , dt}{1 t^4}]

The integral becomes:

[I int frac{2t^2 , dt}{tsqrt{t^4 - 1}}]

Step 3: Partial Fractions

Proceed with partial fractions:

[frac{2t^2}{tsqrt{t^4 - 1}} frac{A}{t - 1} - frac{Bt^2 - 1}{t^4 - 1}]

Solving for (A) and (B), we can proceed to integrate.

Final Integral Solution

The final integral solution is:

[I ln sqrt{sin x} frac{1}{2sqrt{2}} ln left| frac{tan x 1 - sqrt{2 tan x}}{tan x 1 sqrt{2 tan x}} right| C]

This solution combines logarithmic and arctangent terms.

In conclusion, the integral of (sin x cos x) is straightforward and can be computed by recognizing the derivatives of sine and cosine. More complex integrals, such as the one involving (frac{sqrt{sin x}}{sqrt{sin x}sqrt{cos x}}), require clever substitutions and partial fractions.