Problem 13 - Calculus Early Transcendentals

Question

Use the given substitution to find the following indefinite integrals. Check your answer by differentiating. $$\int 2 x\left(x^{2}+1\right)^{4} d x, u=x^{2}+1$$

Step-by-Step Solution

Answer

#Question# Find the indefinite integral of the given function using the substitution provided and verify the answer by differentiating the result: \(\int 2x(x^2 + 1)^4 dx\), with the substitution \(u = x^2 + 1\). #Answer# The indefinite integral of \(\int 2x(x^2 + 1)^4 dx\) is \(\frac{(x^2 + 1)^5}{5} + C\).

Step 1: Identify the expression to replace and its derivative

We are given the substitution $$u = x^2 + 1$$. Let's find the derivative of this expression with respect to x, so that we can replace the \(dx\) term in the integral. Differentiating \(u\) with respect to \(x\), we get: $$\frac{du}{dx} = 2x$$ Now we can rearrange the above equation to get: $$du = 2x \, dx$$

Step 2: Replace the expressions in the integral

Now, we will substitute the expression for \(u\) and \(du\) in the given integral: $$\int 2x (x^{2}+1)^{4}dx = \int (x^{2}+1)^{4} \cdot 2x \, dx$$ Using the substitution \(u = x^{2} + 1\) and \(du = 2x\, dx\), we rewrite the integral as follows: $$= \int u^{4} \, du$$

Step 3: Integrate with respect to u

Now, we can find the integral of \(u^4\) with respect to \(u\): $$\int u^{4} \, du = \frac{u^5}{5} + C$$

Step 4: Replace u with the original expression

Since we used substitution, we need to write our final answer in terms of x. Replace u with the original expression: $$\frac{u^5}{5} + C = \frac{(x^2 + 1)^5}{5} + C$$ So, the indefinite integral of \(\int 2x (x^2+1)^4 dx\) is: $$\frac{(x^2 + 1)^5}{5} + C$$

Step 5: Check the answer by differentiating

To verify our answer, we can differentiate the result with respect to \(x\): $$\frac{d}{dx}\left(\frac{(x^2 + 1)^5}{5} + C\right)$$ Using the chain rule: $$= 5 \cdot (x^2 + 1)^4 \cdot \frac{d}{dx}(x^2 + 1)$$ The derivative of \((x^2 + 1)\) is \(2x\), so the expression becomes: $$= 2x\cdot(x^2 + 1)^4$$ This is the same as the original expression, confirming the correctness of the solution: $$\frac{(x^2 + 1)^5}{5} + C$$

Key Concepts

Indefinite Integrals

Indefinite integrals are essentially the reverse process of differentiation. When calculating an indefinite integral, we're searching for the original function whose derivative gives us the function inside the integral.
It's important to remember that when we integrate, we add a constant denoted as \(+ C\). This constant captures any vertical shift in the original function which wouldn't be visible when differentiating.
This is why, after integrating, the result is expressed as a family of functions, all differing by that constant. The notation for an indefinite integral of a function \(f(x)\) is \(\int f(x) \, dx\). In our exercise, integrating \(2x(x^2+1)^4\) indirectly through substitution led us to \(\frac{(x^2 + 1)^5}{5} + C\).

Chain Rule

The chain rule is a fundamental concept in calculus that makes differentiating composite functions possible.It states that to differentiate a composite function, you multiply the derivative of the outer function by the derivative of the inner function.
This is vital for checking our answers in substitution problems.In our solution, after integrating by substitution, we verified our solution by using the chain rule in differentiation.
To differentiate \(\frac{(x^2 + 1)^5}{5} + C\), we took the outside derivative first.This resulted in multiplying by the derivative of \((x^2 + 1)^5\), which brings in the inner derivative \(2x\).
This returns us to our original integrand \(2x(x^2+1)^4\), confirming the correctness of the procedure.

Differentiation

Differentiation is the process of finding the derivative of a function.A derivative gives us the rate of change of the function with respect to its variable.
In this exercise, we knew we had the correct indefinite integral when we differentiated our result and obtained the original expression.Differentiating our integrated function \(\frac{(x^2 + 1)^5}{5} + C\) brought out the original factors by implementing the chain rule.
The derivative operation reversed the integration step, and this verification is an excellent way to check your work!Whenever possible, differentiate back to reassure that the integration was performed correctly.This keeps your understanding and results precise.

Integration by Substitution

Integration by substitution is a technique used to simplify complex integrals.This involves choosing a substitution that makes the integral easier to solve.
We usually replace a complicated part of the integral with a new variable, like \(u\), which simplifies integration.For our exercise, we set \(u = x^2 + 1\) simplifying the integral to \(\int u^{4} \, du\).
The trick is to transform both the expression and the differential \(dx\) to \(du\). This adjustment allows the integral to take a simpler form which can be easily solved.Once integrated, we revert \(u\) back to the original variable, ensuring the solution is in the correct form for the problem context.Integration by substitution is particularly useful for integrals involving products and powers of functions.