Problem 9
Question
Find the indefinite integral. $$\int\left(x^{2}-1\right)^{9} x d x$$
Step-by-Step Solution
Verified Answer
The indefinite integral of the given function is:
\(\int\left(x^2 - 1\right)^9 x dx = \cfrac{1}{20}(x^2 - 1)^{10} + C\)
1Step1: Choose the substitution function
Let's choose the substitution function \(u(x)\) as follows:
\(u = x^2 - 1\)
2Step2: Find the derivative of u
Now, we can find the derivative du/dx:
\( \cfrac{du}{dx} = 2x\)
3Step3: Find the expression for dx and substitute
Next, we can find an expression for dx, and substitute it into the original integral:
\(dx = \cfrac{du}{2x}\)
Substitute u and dx:
\(\int\left(x^2 - 1\right)^9x dx = \int u^9 \cdot \cfrac{du}{2}\)
4Step4: Simplify and integrate
Now, we can simplify the integral:
\(\int u^9 \cdot \cfrac{du}{2} = \cfrac{1}{2}\int u^9du\)
Now, we can integrate the function:
\(\cfrac{1}{2}\int u^9 du = \cfrac{1}{2} \cfrac{u^{10}}{10} + C\)
5Step5: Substitute back and simplify
Finally, substitute back the original function \(u = x^2 - 1\):
\(\cfrac{1}{2} \cfrac{u^{10}}{10} + C = \cfrac{1}{20}(x^2 - 1)^{10} + C\)
The indefinite integral of the given function is:
\(\int\left(x^2 - 1\right)^9 x dx = \cfrac{1}{20}(x^2 - 1)^{10} + C\)
Key Concepts
Integration TechniquesSubstitution MethodMathematical Problem Solving
Integration Techniques
To tackle integrals efficiently, it's essential to understand various integration techniques available to us. Choosing the correct method can simplify the process significantly. In this particular exercise, we are looking at an indefinite integral. Indefinite integrals essentially involve finding a general form of function whose derivative is equal to the given integral.
One common technique used is **substitution**, which is especially useful when dealing with complicated expressions. It involves breaking down the integral into simpler parts that are easier to integrate. Using substitution helps in transforming a complex-looking integral into a more manageable one.
In this case, the original integral is \( \int (x^2-1)^9 x \, dx \), which, at first glance, might seem tricky to solve directly. With the right technique, such as substitution, it becomes much simpler to approach.
One common technique used is **substitution**, which is especially useful when dealing with complicated expressions. It involves breaking down the integral into simpler parts that are easier to integrate. Using substitution helps in transforming a complex-looking integral into a more manageable one.
In this case, the original integral is \( \int (x^2-1)^9 x \, dx \), which, at first glance, might seem tricky to solve directly. With the right technique, such as substitution, it becomes much simpler to approach.
Substitution Method
The substitution method involves changing the variable of integration to make the integral easier to solve. It is an effective method when you encounter integrals with compositions of functions, like powers or products involving different functions.
First, identify a portion of the expression that can serve as a new variable. In this exercise, we chose to substitute \( u = x^2 - 1 \). This choice simplifies the expression \( (x^2 - 1)^9 \) into a simpler form of \( u^9 \).
Next, find the derivative of the substitution (\( du/dx = 2x \)) to express \( dx \) in terms of \( du \). This step is crucial because it allows you to restate \( dx \) in terms of \( du \), making the integral entirely in terms of \( u \). The relation \( dx = du/(2x) \) is substituted into the integral, changing the original integral to \( \int u^9 \cdot \frac{du}{2} \).
Finally, the integral transforms to a simpler form, and you can solve it using basic integration rules.
First, identify a portion of the expression that can serve as a new variable. In this exercise, we chose to substitute \( u = x^2 - 1 \). This choice simplifies the expression \( (x^2 - 1)^9 \) into a simpler form of \( u^9 \).
Next, find the derivative of the substitution (\( du/dx = 2x \)) to express \( dx \) in terms of \( du \). This step is crucial because it allows you to restate \( dx \) in terms of \( du \), making the integral entirely in terms of \( u \). The relation \( dx = du/(2x) \) is substituted into the integral, changing the original integral to \( \int u^9 \cdot \frac{du}{2} \).
Finally, the integral transforms to a simpler form, and you can solve it using basic integration rules.
Mathematical Problem Solving
In mathematics, solving problems effectively requires not just following steps but understanding the logic behind them. Problem-solving in math is like piecing together a puzzle where understanding each component helps solve the bigger picture.
For the given integral \( \int (x^2-1)^9 x \, dx \), we broke it down into simpler parts using substitution. Recognizing patterns and choosing efficient methods like substitution enhances our ability to solve complex problems.
Ultimately, refining your problem-solving skills comes with practice, understanding, and choosing the techniques that best simplify your calculations.
For the given integral \( \int (x^2-1)^9 x \, dx \), we broke it down into simpler parts using substitution. Recognizing patterns and choosing efficient methods like substitution enhances our ability to solve complex problems.
- Choosing the right method: Opt for substitution when you see nested functions or products like here.
- Manipulating expressions: Turn the problem into a format you are familiar with solving, such as \( u^9 \).
Ultimately, refining your problem-solving skills comes with practice, understanding, and choosing the techniques that best simplify your calculations.
Other exercises in this chapter
Problem 9
Evaluate the definite integral. $$\int_{1}^{2}(2 x-1)^{4} d x$$
View solution Problem 9
Find the area of the region under the graph of the function \(f\) on the interval \([a, b]\). $f(x)=\frac{1}{x} ;[1,2]$$
View solution Problem 9
Find the indefinite integral. $$\int 6 d x$$
View solution Problem 10
Camille purchased a \(15-\mathrm{yr}\) franchise for a computer outlet store that is expected to generate income at the rate of $$ R(t)=400,000 $$ dollars/year.
View solution