Question

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

Short answer

Question: Evaluate the indefinite integral: \(\int 2x(x^2+1)^4 dx\), using the substitution \(u=x^2 + 1\). Then, check your answer by differentiation. Answer: The indefinite integral \(\int 2x(x^2+1)^4 dx\) evaluates to \(\frac{(x^2+1)^5}{5}+C\), and the derivative of that expression confirms its validity.

Step-by-step solution

Introduce the substitution and find the derivative of u

Given our substitution \(u=x^2 + 1\), let's determine \(du\). Differentiating with respect to \(x\): $$\frac{d}{d x}u=\frac{d}{d x}(x^2+1)$$ $$\frac{d}{d x}u=2 x dx$$ Now, let's solve for \(dx\): $$dx=\frac{du}{2x}$$

Replace the variables in the integral using the substitution

Now that we have \(dx\) and the substitution \(u\), we can rewrite the integral in terms of \(u\): $$\int 2x (x^2+1)^4 dx = \int 2x (u)^4 \frac{du}{2x}$$ As you can see, the \(2x\) terms in the numerator and denominator cancel out, leaving us with the integral: $$\int u^4 du$$

Evaluate the integral

Now, evaluate the integral: $$\int u^4 du=\frac{u^5}{5}+C$$ Since we are required to find the indefinite integral in terms of\(x\), we need to rewrite this expression using the definition of \(u\): $$\frac{u^5}{5}+C=\frac{(x^2+1)^5}{5}+C$$

Check by differentiating

Let's check the answer by finding its derivative. The derivative should match the original integrand: $$\frac{d}{d x}\left(\frac{(x^2+1)^5}{5}+C\right)=\frac{5}{5}(x^2+1)^4(2x)=2x(x^2+1)^4$$ Since the result matches the original integrand, our integration is correct. The solution to the given integral is: $$\int 2x(x^2+1)^4 dx = \frac{(x^2+1)^5}{5}+C$$

Key concepts

U-Substitution

The method of u-substitution is a powerful technique for solving indefinite integrals, especially when a composite function is involved.
Think of it as a way to simplify the integral by temporarily swapping out the more complex parts of the function with a single variable, usually denoted as u. This makes the integral look more standard and easier to manage.

Steps to Apply U-Substitution:

• Identify a portion of the integrand that can be substituted with a single variable u.
• Determine du by differentiating u with respect to x.
• Replace all instances of the chosen part with u, and dx with du, making sure the integral is in terms of u only.
• Solve the simplified integral as you would any basic integral.
• After integrating, substitute back the original variables to return to the original variable x.

By applying this method, we can turn a challenging integral into something more familiar and straightforward to solve.

Integration by Substitution

Similar to u-substitution, integration by substitution is useful when integrating a function that is the product of a function and its derivative, or when the integrand is a composite function.
This technique requires some intuition to spot parts of the integrand that can be replaced, simplifying the integral. In essence, it transforms the integral into a simpler form that we can easily tackle.

Typical Scenarios for Integration by Substitution:

• A function is multiplied by its derivative.
• The integral involves a composite function where a function is nested inside another.

After performing the substitution and solving the integral, don't forget to substitute the original variables back in to express the answer in terms of the original variable.

Antiderivative

The concept of the antiderivative is central to integration. An antiderivative of a function f(x) is simply another function F(x) such that when differentiated, it results back in f(x).
Finding an antiderivative means reversing the process of differentiation. For every function f(x), there exists an infinite number of antiderivatives, and these form the family of functions F(x) + C, where C is the constant of integration.

Why Antiderivatives Matter:

• They are integral to solving indefinite integrals.
• Antiderivatives help us calculate the area under a curve in a graph.
• They are used to describe accumulated quantities, like distance or mass.

C represents any potential starting point of accumulation and therefore must be included until additional conditions can specify its value.

Differentiation to Check Integration

One of the best ways to verify that an indefinite integral has been solved correctly is through differentiation. Since taking the derivative of an antiderivative should yield the original function, we can use this property as a way to check our work.
After solving an integral, if we differentiate the result and it matches the original integrand, this confirms the integration process was accurate.

Using Differentiation to Check Your Work:

• Integrate the function to find the antiderivative.
• Use differentiation rules to find the derivative of the antiderivative.
• Compare the derivative with the original integrand. If they are identical, the integration is correct.

This process helps ensure accuracy and builds understanding of the relationship between differentiation and integration.