Problem 22
Question
In Problems 15-34, use the method of substitution to find each of the following indefinite integrals. \(\int x^{2}\left(x^{3}+5\right)^{9} d x\)
Step-by-Step Solution
Verified Answer
The integral of \( x^2(x^3+5)^9 dx \) is \( \frac{(x^3+5)^{10}}{30} + C \).
1Step 1: Choose Substitution Variable
Identify a part of the integrand that can be substituted with a single variable. Here, we choose to let \( u = x^3 + 5 \) because it is inside the parentheses and raised to a power.
2Step 2: Differentiate Substitution Variable
Differentiate the substitution variable with respect to \( x \). Thus, we have \( \frac{du}{dx} = 3x^2 \). Rearranging gives \( du = 3x^2 \, dx \).
3Step 3: Solve for dx
We need to express \( dx \) in terms of \( du \) and \( x \). From \( du = 3x^2 \, dx \), we solve for \( dx \): \( dx = \frac{du}{3x^2} \).
4Step 4: Substitute in Terms of u
Substitute \( u \) and \( dx \) into the integral. The original integral becomes: \( \int x^2 (u)^9 \frac{du}{3x^2} \).
5Step 5: Simplify and Integrate
The \( x^2 \) terms cancel, making the integral \( \int \frac{1}{3} u^9 \, du \). Now, factor out the constant and integrate: \( \frac{1}{3} \int u^9 \, du = \frac{1}{3} \cdot \frac{u^{10}}{10} + C = \frac{u^{10}}{30} + C \).
6Step 6: Substitute u Back in Terms of x
Replace \( u \) with \( x^3 + 5 \) to return to the original variable: \( \frac{(x^3 + 5)^{10}}{30} + C \).
Key Concepts
Substitution MethodIntegral CalculusAlgebraic Manipulation
Substitution Method
The substitution method is a powerful technique in calculus used to simplify the process of integrating functions. When dealing with complex integrals, this method allows us to replace a part of the integral with a single variable, making it easier to solve. The main idea is to identify a function within the integrand that, when substituted, will reduce the complexity of the integral.
For example, in the integral \( \int x^{2}\left(x^{3}+5\right)^{9} \, dx \), the expression \( x^3 + 5 \) is complex, especially since it is raised to a power. By setting \( u = x^3 + 5 \), we simplify the integral because now the expression within the integral becomes easier to handle.
This process involves several steps:
For example, in the integral \( \int x^{2}\left(x^{3}+5\right)^{9} \, dx \), the expression \( x^3 + 5 \) is complex, especially since it is raised to a power. By setting \( u = x^3 + 5 \), we simplify the integral because now the expression within the integral becomes easier to handle.
This process involves several steps:
- Identify and define a substitution variable \( u \).
- Differentially transform \( dx \) in terms of \( du \).
- Substitute both \( u \) and \( dx \) into the integral.
Integral Calculus
Integral calculus is a branch of calculus concerned with the concept of integration, which essentially reverses the process of differentiation. The aim is to determine the accumulation of quantities, which can help in calculating areas, volumes, and other related concepts.
Indefinite integrals, specifically, are integrals without specified bounds. It means solving for the general form of the function and typically includes a constant of integration \( C \). In the provided problem, we end up finding the indefinite integral of a transformed function \( \int u^9 \, du \).
By performing the integration, we obtain \( \frac{u^{10}}{10} + C \). Here we've successfully determined the antiderivative of the function with respect to \( u \). This antiderivative represents a family of functions that describe the original rate of change, including a constant \( C \) that accounts for any shifts up or down the graph of these functions when graphed.
Indefinite integrals, specifically, are integrals without specified bounds. It means solving for the general form of the function and typically includes a constant of integration \( C \). In the provided problem, we end up finding the indefinite integral of a transformed function \( \int u^9 \, du \).
By performing the integration, we obtain \( \frac{u^{10}}{10} + C \). Here we've successfully determined the antiderivative of the function with respect to \( u \). This antiderivative represents a family of functions that describe the original rate of change, including a constant \( C \) that accounts for any shifts up or down the graph of these functions when graphed.
Algebraic Manipulation
Algebraic manipulation is a key technique used throughout mathematics to simplify expressions and solve equations. In the context of substitution and integration, algebraic manipulation allows you to manage the equations efficiently by isolating and rearranging terms.
During the substitution step, after choosing \( u = x^3 + 5 \) and differentiating it to find \( du = 3x^2 \, dx \), algebraic manipulation helps in isolating \( dx \) in terms of \( du \). This process yields \( dx = \frac{du}{3x^2} \).
It crucially allows for elimination of terms and simplification within the integral itself:
During the substitution step, after choosing \( u = x^3 + 5 \) and differentiating it to find \( du = 3x^2 \, dx \), algebraic manipulation helps in isolating \( dx \) in terms of \( du \). This process yields \( dx = \frac{du}{3x^2} \).
It crucially allows for elimination of terms and simplification within the integral itself:
- The term \( x^2 \) cancels out when the expressions are substituted back into the integral.
- Factoring out constants to simplify the integration process, such as factoring out \( \frac{1}{3} \), streamlines the antiderivative computation.
Other exercises in this chapter
Problem 21
In Problems 17-26, find \(G^{\prime}(x)\). $$ G(x)=\int_{x}^{\pi / 4}(s-2) \cot 2 s d s ; 0
View solution Problem 22
In Problems 17-22, calculate \(\int_{a}^{b} f(x) d x\), where a and \(b\) are the left and right end points for which \(f\) is defined, by using the Interval Ad
View solution Problem 22
\(\quad g(y)=\cos 2 y ; \quad[0, \pi]\)
View solution Problem 22
If a function \(f\) is concave down on \([a, b]\), will the Trapezoidal Rule approximation be larger or smaller than \(\int_{a}^{b} f(x) d x\) ?
View solution