Problem 50 - Calculus Early Transcendentals

Question

Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. $$\int\left(a^{2}-t^{2}\right)^{-2} d t$$

Step-by-Step Solution

Answer

Question: Find the indefinite integral of the function \(f(t) = (a^2 - t^2)^{-2}\). Answer: The indefinite integral of the given function is: \(\int\left(a^{2}-t^{2}\right)^{-2} dt = \frac{t}{2a^2(1-t^2/a^2)^{1/2}} + C\), where \(C\) is the constant of integration.

Step 1: Performing Substitution

Let's use the substitution method by setting \(x = t/a\). Now, we can rewrite the integral as: $$\int\left(a^{2}-t^{2}\right)^{-2} dt$$ Substituting \(t = ax\), we get: $$\int\left(a^2 - (ax)^2\right)^{-2} a dx$$ Now, simplifying the function: $$\int\left(a^2(1 - x^2)\right)^{-2} a dx$$ Now, let's denote the differential operator in terms of \(x\). $$\int\left(a^2(1 - x^2)\right)^{-2} a dx$$

Step 2: Integration

We now need to find the integral: $$\int\left(a^2(1 - x^2)\right)^{-2} a dx$$ Factor out the constants: $$a\int(1 - x^2)^{-2} dx$$ Now we can perform the integral using a known result: $$\int(1 - x^2)^{-2} dx = \frac{1}{2a} \frac{x}{(1-x^2)^{1/2}} + C$$

Step 3: Substituting Back

We need to substitute back to get the result in terms of \(t\): $$\frac{1}{2a} \frac{x}{(1-x^2)^{1/2}} + C$$ Substituting \(x = t/a\): $$\frac{1}{2a} \frac{t/a}{(1-(t/a)^2)^{1/2}} + C$$ Now we can simplify the expression: $$\frac{t}{2a^2(1-t^2/a^2)^{1/2}} + C$$

Step 4: Final Answer

The indefinite integral of the given function is: $$\int\left(a^{2}-t^{2}\right)^{-2} dt = \frac{t}{2a^2(1-t^2/a^2)^{1/2}} + C$$ where \(C\) is the constant of integration.

Key Concepts

Substitution Method

The substitution method is a fundamental technique in calculus when dealing with integrals that are not straightforward to solve. It involves transforming the original variable into a new variable, making the integral easier to evaluate. For the integral \( \int (a^2 - t^2)^{-2} \, dt \), we use a substitution method by letting \( x = t/a \). This transforms our integral into \( \int (a^2(1-x^2))^{-2} a \, dx \). Here's how substitution simplifies the problem:

• Substitute \( t = ax \) and thus \( dt = a \, dx \).
• Rewrite the integral: this simplifies the form by factoring out the constants, concentrating the complexity within a simpler algebraic function.
• Evaluate the easier integral form before reverting to the original variable.

This method is particularly useful when a problem contains complex expressions that can be simplified through a change of variables. By carefully choosing the new variable, the substitution method transforms a difficult integral into one that is manageable.

Integration Techniques

Integration techniques offer a myriad of strategies to tackle various forms of integrals, assisting in deriving their antiderivatives. In the given problem, after using substitution, we employ a different technique to resolve the integral of the form \( \int (1-x^2)^{-2} \, dx \).Specific techniques involved in tackling such integrals include:

• Recognizing standard forms: Once simplified, the integral resembles a standard form whose antiderivative is known. This can drastically reduce the complexity of the problem-solving process.
• Factoring out constants: While reducing complexity, this technique involves moving constant multipliers outside of the integral sign, as seen with the factor \( a \) in our problem, simplifying the integration process.
• Direct integration: When possible, utilize direct integration for well-known functions or simplified expressions to quickly obtain the antiderivative, as demonstrated when finding \( \int (1-x^2)^{-2} \, dx \).

Proper application of these techniques not only simplifies the problem but ensures accurate calculation of the integral.

Constant of Integration

In the realm of indefinite integrals, the constant of integration \( C \) plays a crucial role as it represents the family of all possible antiderivatives. When finding indefinite integrals, such as \( \int (a^2 - t^2)^{-2} \, dt \), the constant \( C \) acknowledges that without limits for evaluation, the solution includes all vertical translations of the antiderivative curve. Key points about the constant of integration include:

• It expresses the general solution to an indefinite integral, acknowledging an infinite set of possible solutions.
• Its necessity arises because differentiation of a constant is zero, meaning multiple functions could lead to the same derivative.
• In practical applications, \( C \) can be resolved if additional conditions or boundaries are provided, converting the indefinite integral to a definite one.

Always include \( C \) in your final answer for an indefinite integral, as it represents the unknown initial condition of the original function.