Problem 272 - Calculus Volume 1

Question

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int t\left(1-t^{2}\right)^{10} d t $$

Step-by-Step Solution

Answer

The indefinite integral is \(-\frac{1}{22}(1-t^2)^{11} + C\).

Step 1: Identify the Substitution

To solve the integral \( \int t(1-t^2)^{10} \, dt \), we look for a substitution that simplifies the expression. Notice the term \( 1 - t^2 \) raised to the power 10; this suggests using \( u = 1 - t^2 \) as our substitution. This simplifies the expression under the integral sign.

Step 2: Compute the Derivative of Substitution

With \( u = 1 - t^2 \), compute the differential of \( u \). Recall that if \( u = 1 - t^2 \), then \( \frac{du}{dt} = -2t \). Rearrange this to find \( dt \): \( dt = \frac{du}{-2t} \).

Step 3: Substitute and Simplify the Integral

Substitute \( u \) and \( dt \) in the original integral: \( \int t (1-t^2)^{10} \, dt = \int t u^{10} \, \frac{du}{-2t} \). The \( t \) from the numerator and denominator cancel out, simplifying the integral to \( \int -\frac{1}{2}u^{10} \, du \).

Step 4: Integrate with Respect to \( u \)

Now integrate \( \int -\frac{1}{2}u^{10} \, du \). Use the power rule for integration: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). For this integral, it becomes \( -\frac{1}{2} \cdot \frac{u^{11}}{11} + C \) or \( -\frac{1}{22} u^{11} + C \).

Step 5: Substitute Back to Original Variable

Recall that \( u = 1 - t^2 \). Substitute back into the integrated expression: \( -\frac{1}{22} (1 - t^2)^{11} + C \) is the indefinite integral of the original function.

Key Concepts

Indefinite Integral

An indefinite integral is a fundamental concept in calculus. It represents the collection of all antiderivatives of a function. When you see the symbol \( \int \), it signifies integration, which is the reverse process of differentiation. Essentially, when you take the indefinite integral of a function, you are trying to find a new function whose derivative is the original function you started with. This process helps us determine the accumulation of quantities and how they change over time.- Remember, the solution includes a constant \( C \), because differentiation of a constant is zero. Hence, the indefinite integral of \( f(t) \), denoted as \( F(t) \), is given by: \[ \int f(t) \, dt = F(t) + C \]In the exercise, we look at the function \( t(1-t^2)^{10} \) and apply techniques to find its indefinite integral. Keep in mind that the key here is to reverse the effect of differentiation using integration.

Change of Variables

The change of variables technique is a powerful method used in integration, often simplified with substitutions. It's akin to translating complex expressions into simpler forms for easier calculations. When using the change of variables method, you're essentially changing the perspective or the 'variable' with which you are working with. In the given exercise, we noticed a suitable expression to use for this purpose: - We defined \( u = 1 - t^2 \).The choice of \( 1-t^2 \) aids in simplifying the integral since it is raised to a power and can be easily manipulated. By choosing this substitution, it allows us to express the integral in terms of \( u \), making the integration process smoother.- Calculating the derivative of our substitution, \( \frac{du}{dt} = -2t \), is a critical step.- Rearranging gives us \( dt = \frac{du}{-2t} \), which lets us make appropriate replacements within the integral.These transformations help to simplify matters, reducing complex products or quotients into more manageable polynomials or powers.

Substitution Method

The substitution method is closely related to the change of variables technique. It involves replacing a part of an integral with a new variable to make integration simpler. By using substitution, integrals that are initially difficult to solve become easier to handle. The basic steps include:

• Identify a portion of the integral to substitute, simplifying the variable expression.
• Construct a relationship for both the variable \( u \) and its differential \( du \).
• Replace variables in the integral with the newly defined ones.

In the problem at hand:- We placed \( u = 1 - t^2 \), simplifying the structure.- By cancelling out the variable \( t \) from the numerator and denominator (since \( dt = \frac{du}{-2t} \)), we shifted the integral to \( \int -\frac{1}{2}u^{10} \, du \).After integrating with respect to \( u \), remember to substitute back to the original variable \( t \). This brings us back to the context of the problem, ensuring that our solution matches the form of the initial function.This method is valuable in calculus as it can make solving integrals more intuitive and less cumbersome.