Integral of sqroot( 1+cotx(cotx + cosecx)) ? please answer fast. Thnak you w

To solve the integral of the expression involving the square root of \(1 + \cot x (\cot x + \csc x)\), we can start by simplifying the integrand. The expression inside the square root can be rewritten for clarity, which will help us in the integration process.

Breaking Down the Expression

First, let's simplify the term \(1 + \cot x (\cot x + \csc x)\). We know that:

• \(\cot x = \frac{\cos x}{\sin x}\)
• \(\csc x = \frac{1}{\sin x}\)

Substituting these definitions, we have:

\(\cot x + \csc x = \frac{\cos x}{\sin x} + \frac{1}{\sin x} = \frac{\cos x + 1}{\sin x}\)

Now, substituting back into the original expression:

\(1 + \cot x \left(\cot x + \csc x\right) = 1 + \cot x \cdot \frac{\cos x + 1}{\sin x}\)

Now, substituting \(\cot x\) gives us:

\(1 + \frac{\cos x}{\sin x} \cdot \frac{\cos x + 1}{\sin x} = 1 + \frac{\cos^2 x + \cos x}{\sin^2 x}\)

Combining Terms

Next, we can express this as:

\(1 + \frac{\cos^2 x + \cos x}{\sin^2 x} = \frac{\sin^2 x + \cos^2 x + \cos x}{\sin^2 x}\)

Using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), we simplify this to:

\(\frac{1 + \cos x}{\sin^2 x}\)

Final Form of the Integral

Now, we can rewrite our integral:

\(\int \sqrt{1 + \cot x (\cot x + \csc x)} \, dx = \int \sqrt{\frac{1 + \cos x}{\sin^2 x}} \, dx\)

This simplifies to:

\(\int \frac{\sqrt{1 + \cos x}}{\sin x} \, dx\)

Using Trigonometric Identities

To further simplify \(\sqrt{1 + \cos x}\), we can use the identity:

\(\sqrt{1 + \cos x} = \sqrt{2} \cos\left(\frac{x}{2}\right)\)

Thus, the integral becomes:

\(\int \frac{\sqrt{2} \cos\left(\frac{x}{2}\right)}{\sin x} \, dx\)

Substitution for Integration

To evaluate this integral, we can use the substitution \(u = \frac{x}{2}\), which gives \(dx = 2 \, du\) and \(x = 2u\). The integral now transforms to:

\(2\sqrt{2} \int \frac{\cos u}{\sin(2u)} \, du\)

Using the double angle identity \(\sin(2u) = 2 \sin u \cos u\), we can rewrite the integral as:

\(\sqrt{2} \int \frac{1}{\sin u} \, du\)

Final Integration Step

The integral of \(\frac{1}{\sin u}\) is \(-\ln|\csc u + \cot u| + C\). Therefore, substituting back gives us:

\(-\sqrt{2} \ln\left|\csc\left(\frac{x}{2}\right) + \cot\left(\frac{x}{2}\right)\right| + C\

In summary, the integral of \(\sqrt{1 + \cot x (\cot x + \csc x)}\) leads us to a logarithmic expression involving trigonometric functions. This method illustrates how breaking down complex expressions can simplify the integration process significantly.

To solve the integral of the expression \(\sqrt{1 + \cot x (\cot x + \csc x)}\), we first need to simplify the integrand. Let's break this down step by step.

Understanding the Components

We start with the expression inside the square root:

• \(\cot x\) is defined as \(\frac{\cos x}{\sin x}\).
• \(\csc x\) is defined as \(\frac{1}{\sin x}\).

Rewriting the Expression

Now, let's rewrite the term \(\cot x (\cot x + \csc x)\):

• \(\cot x + \csc x = \frac{\cos x}{\sin x} + \frac{1}{\sin x} = \frac{\cos x + 1}{\sin x}\).
• Thus, \(\cot x (\cot x + \csc x) = \cot x \cdot \frac{\cos x + 1}{\sin x} = \frac{\cos x}{\sin x} \cdot \frac{\cos x + 1}{\sin x} = \frac{\cos x (\cos x + 1)}{\sin^2 x}\).

Combining Terms

Now, substituting this back into the square root gives us:

\(\sqrt{1 + \frac{\cos x (\cos x + 1)}{\sin^2 x}}\).

To combine this under a single square root, we can express 1 as \(\frac{\sin^2 x}{\sin^2 x}\):

\(\sqrt{\frac{\sin^2 x + \cos x (\cos x + 1)}{\sin^2 x}} = \frac{\sqrt{\sin^2 x + \cos^2 x + \cos x}}{\sin x}.\)

Using Trigonometric Identities

Recall the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). Therefore, we can simplify further:

\(\sin^2 x + \cos^2 x + \cos x = 1 + \cos x.\)

This leads us to:

\(\sqrt{1 + \cos x} = \sqrt{2 \cos^2\left(\frac{x}{2}\right)} = \sqrt{2} \cos\left(\frac{x}{2}\right).\)

Final Form of the Integral

Now, substituting this back into our expression gives:

\(\frac{\sqrt{2} \cos\left(\frac{x}{2}\right)}{\sin x}.\)

Thus, our integral becomes:

\(\int \frac{\sqrt{2} \cos\left(\frac{x}{2}\right)}{\sin x} \, dx.\)

Evaluating the Integral

This integral can be approached using substitution or integration techniques. However, the integral of \(\frac{\cos\left(\frac{x}{2}\right)}{\sin x}\) can be complex. A common method is to use trigonometric identities or numerical methods if necessary.

In summary, the integral of \(\sqrt{1 + \cot x (\cot x + \csc x)}\) simplifies to a form that can be integrated with further techniques. Depending on your level of study, you might explore integration by parts or substitution methods to evaluate it completely.