Problem 41 - Calculus Early Transcendentals

Question

Evaluate the following integrals. $$\int \frac{x^{2}}{\left(100-x^{2}\right)^{3 / 2}} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the integral ∫(x^2 / (100 - x^2)^(3/2)) dx. Answer: -ln|√(100 - x^2)| + C

Step 1: Substitution

Let \(x = 10\sin(\theta)\). Then, \(dx = 10\cos(\theta) d\theta\). Substituting these expressions into the integral, we have: $$\int\frac{(10\sin\theta)^2}{\left(100-(10\sin\theta)^2\right)^{3/2}}10\cos\theta d\theta$$

Step 2: Simplify the integral

By simplifying the expression inside the integral, we obtain: $$\int\frac{100\sin^2\theta}{\left(100-100\sin^2\theta\right)^{3/2}}10\cos\theta d\theta$$ Divide both numerator and denominator by \(100\), which gives: $$\int\frac{\sin^2\theta}{\left(1-\sin^2\theta\right)^{3/2}}10\cos\theta d\theta$$ Recall that \(1 - \sin^2\theta = \cos^2\theta\), so the integral becomes: $$10\int\frac{\sin^2\theta}{(\cos^2\theta)^{3/2}}\cos\theta d\theta$$

Step 3: Evaluate the integral

Simplify the integral further and observe that it turns into a simple power function of \(\cos\theta\): $$10\int\sin^2\theta \cos^{-1}\theta d\theta$$ Now, let \(u = \cos\theta\), and thus \(du = -\sin\theta d\theta\). We also change the limits of integration by solving for \(\theta\) when \(x = 0\) and \(x = 2\): $$\int_{u(0)}^{u(\arcsin\frac{x}{10})} -10u^{-1} du$$ Integrating with respect to \(u\), we get: $$-10[\ln|u|]_{u(0)}^{u(\arcsin\frac{x}{10})}$$

Step 4: Transform back to \(x\)

To obtain the result in terms of \(x\), substitute as follows: $$-10[\ln|\cos\theta|]_{\theta=0}^{\theta=\arcsin\frac{x}{10}}$$ The final result is: $$-10\left[\ln\left|\cos\left(\arcsin\frac{x}{10}\right)\right| - \ln|\cos(0)|\right]$$ Recalling the identity \(\cos\left(\arcsin(u)\right) = \sqrt{1-u^2}\), we can further simplify this as: $$-10\ln\left|\frac{\sqrt{100-x^2}}{10}\right|$$ Since \(\ln(10)\) is just a constant, we can include it into the constant of integration \(C\). Therefore, the final answer is: $$-\ln\left|\sqrt{100-x^2}\right| + C$$

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals involving square roots and quadratic expressions. The idea is to use trigonometric identities to transform the integrals into simpler forms. In this exercise, we deal with the integral \(\int \frac{x^{2}}{(100-x^{2})^{3/2}} \, dx\). Here, a trigonometric substitution was utilized to transform the variable \(x\).

The substitution \(x = 10 \sin(\theta)\) was chosen because \( x^2\) appears inside the expression \((100-x^2)\), which is perfect for applying the identity \(\sin^2(\theta) + \cos^2(\theta) = 1\). By letting \(x = 10\sin(\theta)\), the differential becomes \(dx = 10\cos(\theta)\, d\theta\).

• Replaces \(x\) with \(10\sin(\theta)\) simplifies the square root, turning the inside into \(100 - 100\sin^2(\theta)\).
• The identity \(1 - \sin^2(\theta) = \cos^2(\theta)\) is used to further simplify expressions.

Trigonometric substitution is particularly useful for integrals containing expressions under the square root, making calculations more tractable, as shown in the steps transformation.

Integral Simplification

After employing trigonometric substitution, the next challenge is to simplify the resulting integral. This step is all about breaking down complex expressions by using algebraic identities and simplifications that you learn in calculus. Once \(x\) is substituted, the new integral has a more manageable form:

\[10\int \frac{\sin^2(\theta)}{(\cos^2(\theta))^{3/2}}\cos(\theta) \, d\theta \]

Using the identity \(\cos^2(\theta)\) simplifies our integral significantly.

• By recognizing that \(\cos^2(\theta)^{3/2}\) simplifies to \(\cos^3(\theta)\), it turns the integral into a straightforward simplification.
• Multiply through by \(\cos(\theta)\) simplifies the exponent of the cosine, and maintains the correctness of the integral.

Using algebraic simplification transforms the complex powers into more familiar forms, like basic polynomial expressions. This skill in simplifying mends closely with integration techniques, leading to a form easier to evaluate.

Natural Logarithm in Integration

Post simplification and integration by substitution, the next focus is on integrating and expressing the answer using the natural logarithm. Natural logarithms show up frequently in the results of integrals, particularly due to integration involving fractional powers and division. Here we encounter this when expressing our integration result:

The integral after substitution and simplification steps ends up being:

\[-10\int u^{-1} du\]

Where \(u = \cos(\theta)\). This integral is solved using the antiderivative property of natural logarithms:

• The antiderivative of \(u^{-1}\) is \( \ln|u|\), thus solving the integral respects this fundamental principle of calculus.
• This results in an expression of the form \(-10[\ln|u|]\), showing the natural log's role in simplifying the solution.

In the final step, when expressing the solution back in terms of \(x\), we encounter \(\ln|\sqrt{100-x^2}|\), with constant integration adjustments. Recognizing when to employ a natural logarithm is crucial in solving integrals that ultimately involve changing inverses or expressing differentials.