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

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Step 1: Begin by identifying the substitution u = csc(x). This substitution simplifies the integral by expressing cot(x) and csc²(x) in terms of u.

Step 2: Compute the derivative of u with respect to x. Since u = csc(x), we know that du/dx = -csc(x)cot(x). Rearrange this to express dx in terms of du: dx = -du / (csc(x)cot(x)).

Step 3: Rewrite the integral ∫(cot(x)csc²(x))dx using the substitution. Replace csc(x) with u and dx with -du / (csc(x)cot(x)). This transforms the integral into ∫(cot(x)u²)(-du / (u cot(x))).

Step 4: Simplify the expression. Notice that cot(x) cancels out, leaving ∫(-u²)du. The integral is now in terms of u, which is easier to evaluate.

Step 5: Integrate -u² with respect to u. The result will be in terms of u, and you can substitute back u = csc(x) to express the final answer in terms of x.

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Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and partial fractions. In this case, substitution is particularly useful as it simplifies the integral by changing variables, making it easier to evaluate.

Trigonometric Functions

Trigonometric functions, such as cotangent (cot) and cosecant (csc), are fundamental in calculus, especially in integrals involving angles. Understanding their relationships and identities is crucial for manipulating and simplifying expressions during integration. For instance, csc²(x) is the derivative of -cot(x), which can be leveraged in the integration process.

Substitution Method

The substitution method is a technique in calculus where a new variable is introduced to simplify the integration process. By letting u = csc(x), the integral can be transformed into a more manageable form. This method is particularly effective when the integrand contains a function and its derivative, allowing for easier evaluation of the integral.

68. Different methodsb. Evaluate ∫(cot x csc² x) dx using the substitution u=cscx.