Problem 115
Question
Evaluate. $$ \int 9 x\left(7 x^{2}+9\right)^{n} d x, \quad n \neq-1 $$
Step-by-Step Solution
Verified Answer
\( \frac{9}{14(n+1)} (7x^2 + 9)^{n+1} + C \)
1Step 1: Identify the Integral Type
The given integral is of the form \( \int u^n \cdot v' \, dx \), which suggests using the substitution method by selecting an appropriate \( u \). The expression inside the parenthesis, \( 7x^2 + 9 \), is a good candidate for substitution because its derivative appears outside the parenthesis.
2Step 2: Choose Substitution
Let \( u = 7x^2 + 9 \). Then, compute the derivative \( \frac{du}{dx} = 14x \). This means \( du = 14x \, dx \). Notice that we have \( 9x \, dx \) in the integral, which is related but not exactly \( du \).
3Step 3: Adjust \( du \) for Substitution
We need to adjust \( du \) such that it matches our integral's terms. We solve for \( x \, dx \) as \( x \, dx = \frac{du}{14} \). Therefore, \( 9x \, dx = \frac{9}{14}du \).
4Step 4: Substitute and Simplify the Integral
Substitute into the integral: \( \int 9x(7x^2 + 9)^n \, dx = \frac{9}{14} \int u^n \, du \). This simplifies the original integral to a basic power function integral.
5Step 5: Integrate
Integrate \( \frac{9}{14} \int u^n \, du \). The result is \( \frac{9}{14} \cdot \frac{u^{n+1}}{n+1} = \frac{9}{14(n+1)} u^{n+1} \).
6Step 6: Substitute Back
Replace \( u \) with \( 7x^2 + 9 \) to get \( \frac{9}{14(n+1)} (7x^2 + 9)^{n+1} \).
7Step 7: Add the Constant of Integration
Include the constant of integration to obtain the final answer: \[ \frac{9}{14(n+1)} (7x^2 + 9)^{n+1} + C \].
Key Concepts
Substitution MethodDefinite and Indefinite IntegralsPower Function Integration
Substitution Method
The substitution method in integral calculus is a handy tool for simplifying complex integrals. The basic idea is to replace a complicated part of the integral with a single variable, making it easier to evaluate.
In our exercise, we seek to evaluate the integral \( \int 9x(7x^2 + 9)^n \, dx \). Here, the expression inside the parenthesis, \( 7x^2 + 9 \), is our candidate for substitution. This choice works because its derivative, when computed, is proportional to the \( x \, dx \) term outside the parenthesis.
We substitute \( u = 7x^2 + 9 \), and find that \( du = 14x \, dx \). Now, we replace \( x \, dx \) with \( \frac{du}{14} \), ensuring our integral transforms appropriately. This approach streamlines the integration process by allowing us to work with a much simpler power function.
In our exercise, we seek to evaluate the integral \( \int 9x(7x^2 + 9)^n \, dx \). Here, the expression inside the parenthesis, \( 7x^2 + 9 \), is our candidate for substitution. This choice works because its derivative, when computed, is proportional to the \( x \, dx \) term outside the parenthesis.
We substitute \( u = 7x^2 + 9 \), and find that \( du = 14x \, dx \). Now, we replace \( x \, dx \) with \( \frac{du}{14} \), ensuring our integral transforms appropriately. This approach streamlines the integration process by allowing us to work with a much simpler power function.
Definite and Indefinite Integrals
Integrals are a fundamental concept in calculus, divided into two main categories: definite and indefinite integrals. An indefinite integral, like the one in our example, is an integral without defined limits. It generally includes an integration constant (denoted as \( C \)) to express the family of possible antiderivatives.
For our integral: \[ \int 9x(7x^2 + 9)^n \, dx \], we do not have specific limits to evaluate; hence, it's an indefinite integral. After performing substitution and integration, we must add \( C \) to represent all potential antiderivatives. This constant is crucial because it accounts for all vertical shifts of the antiderivative function graph.
Unlike definite integrals, which give us a numerical value for an area under a curve, indefinite integrals focus on finding the general form of the antiderivative without specific boundaries.
For our integral: \[ \int 9x(7x^2 + 9)^n \, dx \], we do not have specific limits to evaluate; hence, it's an indefinite integral. After performing substitution and integration, we must add \( C \) to represent all potential antiderivatives. This constant is crucial because it accounts for all vertical shifts of the antiderivative function graph.
Unlike definite integrals, which give us a numerical value for an area under a curve, indefinite integrals focus on finding the general form of the antiderivative without specific boundaries.
Power Function Integration
Power function integration is the process of integrating functions of the form \( x^n \), where \( n \) is a real number. The power rule for integration states that the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} \), provided \( n eq -1 \).
In the example \( \int \frac{9}{14} u^n \, du \), we apply this rule to the variable \( u \). The integration becomes straightforward as we use the power rule to find that \( \int u^n \, du = \frac{u^{n+1}}{n+1} \).
Multiplying this result by \( \frac{9}{14} \) gives us \( \frac{9}{14(n+1)} u^{n+1} \). Finally, when we substitute back \( u = 7x^2 + 9 \), we complete our integration: \( \frac{9}{14(n+1)}(7x^2 + 9)^{n+1} \).
This integration technique is essential for simplifying expressions into a form that's easy to evaluate and understand.
In the example \( \int \frac{9}{14} u^n \, du \), we apply this rule to the variable \( u \). The integration becomes straightforward as we use the power rule to find that \( \int u^n \, du = \frac{u^{n+1}}{n+1} \).
Multiplying this result by \( \frac{9}{14} \) gives us \( \frac{9}{14(n+1)} u^{n+1} \). Finally, when we substitute back \( u = 7x^2 + 9 \), we complete our integration: \( \frac{9}{14(n+1)}(7x^2 + 9)^{n+1} \).
This integration technique is essential for simplifying expressions into a form that's easy to evaluate and understand.
Other exercises in this chapter
Problem 113
Evaluate. $$ \int \frac{(\ln x)^{n}}{x} d x, x>0, \quad n \neq-1 $$
View solution Problem 114
Evaluate. $$ \int \frac{e^{-m x}}{1-a e^{-m x}} d x $$
View solution Problem 116
Evaluate. $$ \int 5 x^{2}\left(2 x^{3}-7\right)^{n} d x, \quad n \neq-1 $$
View solution Problem 117
Is the following a true statement? Why or why not? $$\int[2 f(x)] d x=[f(x)]^{2}+C$$
View solution