https://file.aiquickdraw.com/mathbot/file/1728268542716m182e976.jpg

The integral provided is:

\[
\int \cot^4(x) \csc^3(x) \, dx
\]

Let's work through the steps to solve this integral.

### Step 1: Express everything in terms of \(\csc(x)\) and \(\cot(x)\)
We'll use the identities:

\[
\csc(x) = \frac{1}{\sin(x)} \quad \text{and} \quad \cot(x) = \frac{\cos(x)}{\sin(x)}.
\]

However, for integrals like this, it’s helpful to simplify the powers of \(\cot(x)\) and \(\csc(x)\) and apply trigonometric identities directly.

### Step 2: Apply trigonometric identity
The derivative of \(\csc(x)\) is:

\[
\frac{d}{dx} \left[ \csc(x) \right] = -\csc(x) \cot(x)
\]

This suggests that a good substitution would involve \(\csc(x)\).

### Step 3: Substitution
Let:
\[
u = \csc(x)
\]

Then:
\[
du = -\csc(x) \cot(x) \, dx
\]

Thus, \(\cot(x) \, dx = -\frac{du}{u}\).

We can now express the integral in terms of \(u\). Since \(\cot^4(x) = u^4 - 1\) and \(\csc(x) = u\){,}.
I'll work

Learn how to solve the integral of cot^4(x) · csc^3(x) using trigonometric identities and substitution. This problem covers advanced trigonometric functions and integration techniques, suitable for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation