Question

Trigonometric substitutions Evaluate the following integrals using trigonometric substitution. $$\int \frac{x^{3}}{\left(81-x^{2}\right)^{2}} d x$$

Short answer

Question: Evaluate the integral of the function: \(\int \frac{x^{3}}{\left(81-x^{2}\right)^{2}} d x\) Answer: \(\frac{x^2}{2(81 - x^2)} + C\)

Step-by-step solution

Choose a suitable substitution

One possible trigonometric substitution that simplifies the denominator is a sine substitution: \(x = 9\sin(\theta)\). This works because, according to the Pythagorean identity, \(\sin^2(\theta) + \cos^2(\theta) = 1\). This substitution will turn the \(81 - x^2\) term in the denominator into a more manageable form of \(81\cos^2(\theta)\).

Find the derivative of the substitution

Now that we have our substitution, we'll need to find the derivative with respect to \(\theta\), which will be used to replace the differential \(dx\) in the integral. Differentiate \(x = 9\sin(\theta)\) with respect to \(\theta\): $$\frac{dx}{d\theta} = 9\cos(\theta)$$ Then, $$dx = 9\cos(\theta) d\theta$$

Rewrite the integral using the substitution

Now we will rewrite the integral using our trigonometric substitution and replacing dx with the expression we found in step 2: $$\int \frac{x^{3}}{\left(81-x^{2}\right)^{2}} d x = \int \frac{(9\sin(\theta))^{3}}{(81 - (9\sin(\theta))^{2})^2} (9\cos(\theta) d\theta) = \int \frac{729\sin^3(\theta)\cos(\theta)}{81\cos^4(\theta)}d\theta$$

Simplify the integral

Next, we need to simplify this integral. Cancel out the terms and rewrite the expression: $$\int \frac{9\sin^3(\theta)}{\cos^3(\theta)}d\theta$$

Evaluate the integral

To evaluate this integral, we apply integration by substitution once more using \(u = \cos(\theta)\), then \(-\frac{du}{d\theta} = \sin(\theta)d\theta\) So, the integral becomes $$-\int \frac{1}{u^3}du$$ Now we can solve to get: $$-\int \frac{1}{u^3}du = \frac{1}{2u^2} + C$$ Here C represents the constant of integration.

Substitute back to get the answer

Now, we need to substitute back to get our answer in terms of \(x\): $$\frac{1}{2u^2} + C = \frac{1}{2\cos^2(\theta)} + C$$ Since we had \(x = 9\sin(\theta)\), using Pythagorean identity, we have $$\cos^2(\theta) = \frac{81 - x^2}{81}$$ Thus, the final answer is: $$\frac{1}{2\cos^2(\theta)} + C = \frac{1}{2(\frac{81 - x^2}{81})} + C = \frac{x^2}{2(81 - x^2)} + C$$

Key concepts

Integral Calculus

Integral calculus is a branch of mathematics concerned with the accumulation of quantities and the areas under and between curves. It's used to find quantities such as areas, volumes, and the total accumulation of a function over an interval. In the context of the problem provided, integral calculus allows us to sum up an infinite number of infinitesimally small pieces to calculate a total area or volume.

For instance, in our exercise, we're trying to evaluate the integral of a function involving a quotient with a squared term in the denominator. The integral, in this case, will give us the signed area under the curve of the function \( \frac{x^{3}}{(81-x^{2})^{2}} \), which is not easy to calculate directly due to the squared term in the denominator. This complex expression leads us to seek out integration techniques such as trigonometric substitution to simplify the integral and make it solvable.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. They can be incredibly useful in simplification and solving of trigonometry problems, particularly within integral calculus. One of the most fundamental identities is the Pythagorean identity, \( \text{sin}^{2}(\theta) + \text{cos}^{2}(\theta) = 1 \), which is derived from the Pythagorean theorem applied to a right triangle.

In our exercise, we utilize the substitution \( x = 9\text{sin}(\theta) \) strategically, recognizing that \( 81 - x^2 \) becomes \( 81\text{cos}^2(\theta) \), a much simpler expression to integrate—thanks to the Pythagorean identity. This substitution elegantly transforms a complicated algebraic expression into one that can be approached with standard integration techniques.

Integration Techniques

Integration techniques encompass a variety of methods used to compute complex integrals that are not straightforward. Common techniques include substitution, integration by parts, partial fraction decomposition, and trigonometric substitution among others.

In trigonometric substitution, as demonstrated in the exercise, we make a deliberate choice to replace the variable with a trigonometric function to leverage trigonometric identities for simplifying the integral. After performing the substitution, additional steps are often required to evaluate the new integral. For example, we might need to apply another type of substitution as we did with \( u = \text{cos}(\theta) \) to handle the resulting trigonometric expression, or we might need to use another approach depending on the integrand.

Understanding when and how to apply these techniques is crucial for solving integrals effectively. It requires practice and familiarity with various types of functions and their associated strategies. The ability to recognize which technique to use is a key skill in integral calculus, as illustrated by the considered and methodical approach employed for the given problem.