Question

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. $$ \int_{-1}^{1}\left(1-x^{2}\right)^{3 / 2} d x $$

Short answer

The integral evaluates to \(-\frac{2}{3}\).

Step-by-step solution

Identify the Trigonometric Substitution

The integrand is of the form \((1-x^2)^{3/2}\), which suggests trigonometric substitution. Use the substitution \(x = \sin(\theta)\), thus \(dx = \cos(\theta)\,d\theta\). The limits \(x = -1\) to \(x = 1\) transform to \(\theta = -\frac{\pi}{2}\) to \(\theta = \frac{\pi}{2}\).

Substitute and Simplify the Integral

Substitute \(x = \sin(\theta)\) into the integral:\[\int_{-\pi/2}^{\pi/2} \left(1 - \sin^2(\theta)\right)^{3/2} \cos(\theta)\, d\theta\]This simplifies to:\[\int_{-\pi/2}^{\pi/2} \cos^3(\theta)\, d\theta\]since \(1 - \sin^2(\theta) = \cos^2(\theta)\) and \( (\cos^2(\theta))^{3/2} = \cos^3(\theta)\).

Calculate the Integral

Now integrate \(\cos^3(\theta)\). Use the identity \(\cos^3(\theta) = \cos(\theta) - \cos(\theta) \sin^2(\theta)\), which can be expressed as \(\cos(\theta)(1 - \sin^2(\theta))\). Therefore,\[\int_{-\pi/2}^{\pi/2} (\cos(\theta) - \cos(\theta) \sin^2(\theta))\, d\theta\]This becomes two separate integrals:\[\int_{-\pi/2}^{\pi/2} \cos(\theta)\, d\theta - \int_{-\pi/2}^{\pi/2} \cos(\theta) \sin^2(\theta)\, d\theta\]

Evaluate the Integrals

The first integral \(\int_{-\pi/2}^{\pi/2} \cos(\theta)\, d\theta\) evaluates to zero as it is symmetric around zero. The second integral can be solved using substitution \(u = \sin(\theta)\). Thus, \(du = \cos(\theta)\,d\theta\) and limits from \(-1\) to \(1\):\[\int_{-1}^{1} u^2\, du = \left[\frac{u^3}{3}\right]_{-1}^{1} = \frac{1}{3}(1 - (-1)) = \frac{2}{3}\]Therefore,\(0 - \frac{2}{3} = -\frac{2}{3}\).

Conclude the Integration

The value of the integral \(\int_{-1}^{1} (1-x^2)^{3/2}\,dx\) is \(-\frac{2}{3}\) as obtained from the evaluation of substituted and transformed integrals. This completes the integration process using trigonometric substitution.

Key concepts

Integration Techniques

Integration techniques are strategies used to simplify and solve complex integrals. One very effective strategy is trigonometric substitution. This method is specifically useful for integrals involving square roots, like \(\sqrt{a^2 - x^2}\), \(\sqrt{x^2 + a^2}\), or \(\sqrt{x^2 - a^2}\). Trigonometric substitution simplifies these integrals by using the trigonometric identities to eliminate the square root expressions, replacing them with trigonometric functions.For example, if the integral involves \(1-x^2\), the substitution \(x = \sin(\theta)\) is commonly used because \(1 - \sin^2(\theta) = \cos^2(\theta)\). This transforms the integral into one that involves simpler trigonometric functions that can be more easily integrated. It's essential to adjust the differential \(dx\) also, replacing \(dx\) with \(\cos(\theta) d\theta\). After performing the integration, you must transform back into the original variable using the inverse trigonometric function.

Definite Integrals

Definite integrals have a very special role in calculus, as they represent the area under a curve between two points. When performing trigonometric substitution in a definite integral, like \(\int_{-1}^{1} (1-x^2)^{3/2} \, dx\), it's important to adjust the limits of integration according to the substitution used.In our example, the original limits were \(x = -1\) and \(x = 1\). After substituting \(x = \sin(\theta)\), these limits convert to angles: \(\theta = -\frac{\pi}{2}\) and \(\theta = \frac{\pi}{2}\). Be careful when converting back if the inverse function has different restrictions.Setting up the definite integral with correct limits ensures that the calculus tools calculate the area accurately. It is critical, especially when dealing with symmetry, as shown when \(\int_{-\pi/2}^{\pi/2} \cos(\theta) \, d\theta\) yields zero due to the symmetric properties of cosine over this interval.

Calculus Problem Solving

Calculus problem solving often involves combining different techniques to break down and solve a problem systematically. In our exercise, we used trigonometric substitution and then split the problem into simpler integrals.Once the substitution was made, simplifying \(\left(1 - \sin^2(\theta)\right)^{3/2}\) to \(\cos^3(\theta)\) allowed for integration using trigonometric identities such as \(\cos^3(\theta) = \cos(\theta) - \cos(\theta)\sin^2(\theta)\). This identity separated the integral into parts that could be tackled individually.After breaking the problem into smaller parts, we solved each integral separately. To solve the second integral, the substitution \(u = \sin(\theta)\) was implemented, transforming it into a polynomial integral \(\int u^2 \, du\). Solving this and combining results gives the final value. Each step requires careful manipulation and attention to detail, ensuring all transformations are valid and accurate.