Problem 35 - Calculus: Early Transcendentals

Question

Evaluate the indefinite integral. \( \displaystyle \int \sqrt{\cot x} \csc^2 x \, dx \)

Step-by-Step Solution

Answer

The integral evaluates to \(-\frac{2}{3}(\cot x)^{3/2} + C\)."

Step 1: Identify Substitution

Consider the derivative of some trigonometric identities involved. We know that the derivative of \( \cot x \) is \( -\csc^2 x \). So, we choose \( u = \cot x \).

Step 2: Differentiate u with Respect to x

Differentiate \( u \) to find \( du \). Since \( u = \cot x \), we have \( du = -\csc^2 x \, dx \).

Step 3: Rewrite the Integral in terms of u

Substitute \( u = \cot x \) and \( du = -\csc^2 x \, dx \) into the integral. The integral becomes: \( \int \sqrt{u} \, (-du) = -\int u^{1/2} \, du \).

Step 4: Integrate with respect to u

Use the power rule to integrate. The power rule states \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). So, \( -\int u^{1/2} \, du = -\left( \frac{u^{3/2}}{3/2} \right) + C = -\frac{2}{3}u^{3/2} + C \).

Step 5: Substitute back the original variable

Replace \( u \) with \( \cot x \) to get the answer in terms of \( x \). This gives \(-\frac{2}{3}(\cot x)^{3/2} + C \).

Key Concepts

Understanding Trigonometric Substitution

Trigonometric substitution is a powerful technique used in calculus to simplify integrals that involve trigonometric functions. It leverages the intrinsic relationships between trigonometric identities to transform complex expressions into easier ones.

When evaluating integrals containing expressions like \( \sqrt{\cot x} \csc^2 x \), we often turn to trigonometric substitution for help.

• In our original problem, notice how the integral involves both \( \cot x \) and \( \csc^2 x \).
• Given the trigonometric identity where the derivative of \( \cot x \) is \( -\csc^2 x \), we set \( u = \cot x \).

This substitution simplifies the problem by transforming the trigonometric function into a polynomial form through the subsequent calculation of the differential \( du = -\csc^2 x \, dx \).

Now, we can rewrite the original integral completely in terms of \( u \), making integration possible using different, less complex methods.

Applying the Power Rule for Integration

The power rule of integration is one of the fundamental techniques used to integrate expressions involving power functions. It states that the integral of \( u^n \) with respect to \( u \) is \( \frac{u^{n+1}}{n+1} + C \), where \( n eq -1 \).

In our exercise, after substituting \( u = \cot x \), the integral becomes \( -\int u^{1/2} \ du \).

• Using the power rule, we can find the antiderivative of \( u^{1/2} \), which is \( \frac{u^{3/2}}{3/2} \).
• Subsequently, when accounting for the negative sign, the integrated result is \(-\frac{2}{3}u^{3/2} + C \).

This process showcases how simple rules of integration can transform and solve an integral derived from a more complex trigonometric expression.

Integration by Substitution Explained

Integration by substitution is a critical method in calculus that simplifies the integration process by changing variables. It's often used when a part of the integral's structure resembles a derivative of another function.

In the context of our problem, integration by substitution arises when we substitute \( u = \cot x \), turning the trigonometric functions into a more manageable polynomial form.

• The original differential \( dx \) is transformed using the derivative \( du = -\csc^2 x \, dx \), leading to a neater integral \(-\int u^{1/2} \ du \).
• This change simplifies the integration task from a trigonometric to a power function, which is far easier to handle.

Once the integral is evaluated in terms of \( u \), the solution is then converted back into the original variable with \( u = \cot x \), yielding \(-\frac{2}{3}(\cot x)^{3/2} + C \).

The process of substitution not only simplifies solving complex integrals but also serves as an essential connecting thread between trigonometric identities and basic algebraic processes.