Problem 22 - Calculus Early Transcendentals

Question

Evaluate the following integrals. $$\int \frac{d t}{t^{2} \sqrt{9-t^{2}}}$$

Step-by-Step Solution

Answer

Answer: The value of the integral is \(-\frac{\sqrt{1-\frac{t^2}{9}}}{t} + C\).

Step 1: Apply the trigonometric substitution

Let \(t = 3\sin\theta\). Then, differentiate with respect to \(\theta\) to find the differential \(dt\): $$\frac{d t}{d\theta} = 3\cos\theta$$ $$d t = 3\cos\theta d\theta.$$ Substitute \(t = 3\sin\theta\) and \(d t = 3\cos\theta d\theta\) into the integral: $$\int \frac{3\cos\theta d\theta}{(3\sin\theta)^{2} \sqrt{9-(3\sin\theta)^{2}}}.$$

Step 2: Simplify the integrand

Simplify the expression under the square root and the denominator: $$\int \frac{3\cos\theta d\theta}{9\sin^2\theta \sqrt{9-9\sin^2\theta}} .$$ Factor out a \(9\) from under the square root: $$\int \frac{3\cos\theta d\theta}{9\sin^2\theta \sqrt{1-\sin^2\theta}} .$$ Now, apply the Pythagorean Identity: \(1-\sin^2\theta=\cos^2\theta\) and simplify: $$\int \frac{3\cos\theta d\theta}{9\sin^2\theta \sqrt{\cos^2\theta}} .$$ Take the square root of \(\cos^2\theta\), which is simply \(|\cos\theta|\), and simplify further: $$\int \frac{3\cos\theta d\theta}{9\sin^2\theta |\cos\theta|}.$$ Since we are working in the interval \([-\pi/2, \pi/2]\), the cosine function is positive, and thus we can drop the absolute value: $$\int \frac{3\cos\theta d\theta}{9\sin^2\theta \cos\theta}.$$ Cancel the common factor \(\cos\theta\): $$\int \frac{3 d\theta}{9\sin^2\theta}.$$

Step 3: Integrate and simplify the expression

We can now integrate with respect to \(\theta\): $$\frac{1}{3} \int \frac{d\theta}{\sin^2\theta}.$$ This integral is a well-known integral, the integral of the cosecant squared function. The antiderivative of \(\csc^2\theta\) is \(- \cot\theta\): $$-\frac{1}{3}\cot\theta + C.$$

Step 4: Convert back to the original variable \(t\)

Recall that we started with the substitution \(t = 3\sin\theta\). To find \(\cot\theta\), we can first find \(\cos\theta\) using the Pythagorean Identity: $$\cos^2\theta = 1 - \sin^2\theta$$ $$\cos\theta =\sqrt{1-\frac{t^2}{9}}.$$ Thus, \(\cot\theta=\frac{\cos\theta}{\sin\theta}\), and we can replace \(\sin\theta\) with \(\frac{t}{3}\): $$\cot\theta = \frac{\sqrt{1-\frac{t^2}{9}}}{\frac{t}{3}}.$$ Substitute back into the expression we found earlier: $$-\frac{1}{3}\cdot\frac{3\sqrt{1-\frac{t^2}{9}}}{t} + C.$$ Simplify the expression: $$-\frac{\sqrt{1-\frac{t^2}{9}}}{t} + C.$$

Step 5: Write the final answer

The final result is the following: $$\int \frac{dt}{t^2\sqrt{9-t^2}} = -\frac{\sqrt{1-\frac{t^2}{9}}}{t} + C.$$

Key Concepts

Trigonometric Identity

Trigonometric identities are valuable tools used in calculus to simplify expressions. They often make integration simpler, especially when dealing with complex formulas. In this exercise, one such identity was essential for simplifying the integral.
The identity used is the Pythagorean Identity: \(1 - \sin^2\theta = \cos^2\theta\). This identity allows a transformation, rewriting terms involving sine and cosine to make them more manageable during integration.
By applying this identity, \(\sqrt{9-9\sin^2\theta}\) was simplified to \(\sqrt{9}\sqrt{1 - \sin^2\theta}\), and further to \(3|\cos\theta|\). Since we were evaluating on \([-\pi/2, \pi/2]\), \(|\cos\theta| = \cos\theta\). This simplification was crucial to proceed with the integration steps, making the problem accessible.

Definite Integrals

Definite integrals provide the tool to calculate the area under a curve within specified boundaries. While this exercise involved an indefinite integral, understanding definite integrals helps grasp these processes.
With definite integrals, the goal is to find a precise value that represents the accumulation of quantities over an interval. In practice, this means evaluating the antiderivative at the upper and lower bounds and taking their difference.

• Set up the integral with limits to define the area or quantity.
• Perform the integration process, considering any substitutions or transformations needed.
• Apply the Fundamental Theorem of Calculus, which states that the definite integral can be found using antiderivatives.
• Subtract the lower bound evaluation from the upper bound evaluation.

In the current context, receiving a result expressed as \(-\frac{\sqrt{1-\frac{t^2}{9}}}{t} + C\) means the integration is indefinite. If we wanted an area, definite bounds would be required.

Antiderivatives

Antiderivatives, often used interchangeably with indefinite integrals, are the reverse of differentiation. They help in determining an original function from its derivative, providing powerful insights into calculus problems.
The antiderivative process here involved recognizing that the integral of \(\csc^2\theta\) is \(-\cot\theta\). Knowing common antiderivatives, like this, speeds up finding solutions.

• The goal is transforming the integral into one of simpler or recognizable form.
• Apply previously known results or identities to reach the standard form.
• Integrate using standard rules and substitution if necessary.

In this problem, the integral simplified to \(-\frac{1}{3} \cot\theta + C\). Returning this function to the original variable \(t\) required understanding the substitution’s impact on the function. Thus, grasping antiderivatives is pivotal in solving such integrals efficiently.