68. Different methods
a. Evaluate ∫(cot x csc² x) dx using the substitution u=cotx.

Verified step by step guidance

1. Step 1: Identify the substitution. Let u = cot(x). This substitution simplifies the integral by replacing cot(x) with u.
2. Step 2: Compute the derivative of u with respect to x. Since u = cot(x), we know that du/dx = -csc²(x). Rearrange this to express dx in terms of du: dx = -du/csc²(x).
3. Step 3: Rewrite the integral in terms of u. Substitute u = cot(x) and dx = -du/csc²(x) into the integral ∫(cot(x) csc²(x)) dx. This gives ∫(u csc²(x) * (-du/csc²(x))).
4. Step 4: Simplify the integral. Notice that csc²(x) cancels out, leaving ∫(-u du). This simplifies the integral to -∫u du.
5. Step 5: Integrate with respect to u. The integral of u with respect to u is (u²/2), so the result becomes -(u²/2). Finally, substitute back u = cot(x) to express the solution in terms of x: -(cot²(x)/2).

Key Concepts

Integration

Integration is a fundamental concept in calculus that involves finding the integral of a function, which represents the area under the curve of that function. It can be thought of as the reverse process of differentiation. In this context, we are tasked with evaluating an integral, which requires understanding how to manipulate and simplify expressions to find their antiderivatives.

Substitution Method

The substitution method is a technique used in integration to simplify the process by changing variables. By substituting a new variable (in this case, u = cot x), we can transform the integral into a more manageable form. This method is particularly useful when dealing with composite functions, allowing us to express the integral in terms of a single variable.

Trigonometric Functions

Trigonometric functions, such as cotangent (cot) and cosecant (csc), are essential in calculus, especially in integrals involving angles. Understanding their properties and relationships is crucial for manipulating expressions involving these functions. In this problem, recognizing how cot x and csc² x relate to each other will aid in simplifying the integral after substitution.