Problem 273
Question
In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int(11 x-7)^{-3} d x $$
Step-by-Step Solution
Verified Answer
\(-\frac{1}{22(11x -7)^2} + C\).
1Step 1: Identify the Variable to Substitute
In the integral \( \int(11x-7)^{-3} \, dx \), the expression \(11x-7\) is inside the integrand. A suitable substitution is \( u = 11x - 7 \).
2Step 2: Differentiate to Find \( du \)
Differentiate \( u = 11x - 7 \) with respect to \( x \) to find \( du \). We have \( \frac{du}{dx} = 11 \), so \( du = 11 \, dx \).
3Step 3: Solve for \( dx \) in Terms of \( du \)
From \( du = 11 \, dx \), we solve for \( dx \) which gives \( dx = \frac{du}{11} \).
4Step 4: Substitute \( u \) and \( dx \) in the Integral
Replace \( 11x-7 \) with \( u \) and \( dx \) with \( \frac{du}{11} \) in the integral. The integral becomes: \[ \int u^{-3} \cdot \frac{du}{11} = \frac{1}{11} \int u^{-3} \, du. \]
5Step 5: Integrate with Respect to \( u \)
The integral \( \int u^{-3} \, du \) can be integrated using power rule: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). Here, \( n = -3 \), so the integral becomes \(-\frac{1}{2}u^{-2} \). Therefore, \[ \frac{1}{11} \int u^{-3} \, du = \frac{1}{11} \left( -\frac{1}{2} u^{-2} \right) + C. \]
6Step 6: Simplify the Integral
Simplify the expression: \(-\frac{1}{22} u^{-2} + C \) which gives \(-\frac{1}{22u^2} + C \).
7Step 7: Back-Substitute \( u \) in Terms of \( x \)
Replace \( u \) with the original expression \( 11x-7 \) to get the indefinite integral in terms of \( x \): \(-\frac{1}{22(11x -7)^2} + C \).
Key Concepts
Change of VariablesSubstitution MethodPower Rule for Integration
Change of Variables
Change of variables is a fundamental concept in integration and calculus. It helps simplify complex integrals by transforming variables into ones that are easier to integrate. In the process, we replace the original variable of integration with a new variable that usually simplifies the expression inside the integral.
By changing variables, we often make an integrand, the function being integrated, more manageable. This technique makes it easier to apply integration rules and calculations.
In the integral \( \int(11x-7)^{-3} \, dx \), we observe that \(11x - 7\) is a linear expression. Replacing this with the variable \( u \) changes the scope of the integral from \( x \) to \( u \), simplifying the function we need to integrate. Overall, change of variables is a clever trick to make seemingly complex integrals straightforward.
By changing variables, we often make an integrand, the function being integrated, more manageable. This technique makes it easier to apply integration rules and calculations.
In the integral \( \int(11x-7)^{-3} \, dx \), we observe that \(11x - 7\) is a linear expression. Replacing this with the variable \( u \) changes the scope of the integral from \( x \) to \( u \), simplifying the function we need to integrate. Overall, change of variables is a clever trick to make seemingly complex integrals straightforward.
Substitution Method
The substitution method is a specific technique in calculus used to perform integrations involving complex expressions. The main idea is to take part of the integrand and substitute it with a new variable (usually \( u \)), making the integral easier to solve.
For instance, if we tackle the integral \( \int(11x-7)^{-3} \, dx \), the substitution \( u = 11x - 7 \) simplifies the process. By substituting \( u \), we transform the integral into one involving \( u \) instead of \( x \).
Steps involved in substitution are:
For instance, if we tackle the integral \( \int(11x-7)^{-3} \, dx \), the substitution \( u = 11x - 7 \) simplifies the process. By substituting \( u \), we transform the integral into one involving \( u \) instead of \( x \).
Steps involved in substitution are:
- Identify a part of the integrand to substitute with \( u \).
- Differentiate \( u \) to find \( du \) in terms of \( dx \).
- Rearrange the expression to solve for \( dx \).
- Modify the integral to only involve \( u \) and \( du \).
- Finally, perform the integration and back-substitute \( u \) with the original expression.
Power Rule for Integration
The power rule for integration is an essential formula for finding antiderivatives of functions. It states that to integrate a power of \( u \), you can use: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \), where \( C \) is the constant of integration and \( n eq -1 \).
In our example, once we've performed substitution and simplified the integral to \( \int u^{-3} \, du \), the power rule helps us compute the integral. When \( n = -3 \), the calculation yields \( -\frac{1}{2} u^{-2} + C \), which becomes the antiderivative.
This rule is simple yet powerful, making it a vital tool in integration problems. However, always be cautious to adjust \( n \) appropriately and apply the rule only when suitable.
In our example, once we've performed substitution and simplified the integral to \( \int u^{-3} \, du \), the power rule helps us compute the integral. When \( n = -3 \), the calculation yields \( -\frac{1}{2} u^{-2} + C \), which becomes the antiderivative.
This rule is simple yet powerful, making it a vital tool in integration problems. However, always be cautious to adjust \( n \) appropriately and apply the rule only when suitable.
Other exercises in this chapter
Problem 271
In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int x(1-x)^{99} d x $$
View solution Problem 272
In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int t\left(1-t^{2}\right)^{10} d t $$
View solution Problem 274
In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int(7 x-11)^{4} d x $$
View solution Problem 275
In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int \cos ^{3} \theta \sin \theta d \theta $$
View solution