Problem 11 - Calculus Single Variable

Question

Use the method of substitution to calculate the indefinite integrals. $$ \int 16 x\left(x^{2}+1\right)^{7} d x $$

Step-by-Step Solution

Answer

The integral is \( (x^2+1)^8 + C \).

Step 1: Choice of Substitution

The given integral is \( \int 16x(x^2+1)^7 \, dx \). We notice that \( x^2+1 \) is a composite function, so it's suitable for substitution. Let's set \( u = x^2 + 1 \).

Step 2: Compute the Derivative (du)

To find \( du \), differentiate \( u = x^2 + 1 \) with respect to \( x \). This gives \( du = 2x \, dx \). For substitution, we need \( 16x \, dx \), so let's express this in terms of \( du \): \( 16x \, dx = 8 \, du \).

Step 3: Substitute and Change Limits

Using the substitution, the integral becomes \( \int 8(u)^7 \, du \) after replacing \( 16x \, dx \) with \( 8 \, du \) and \( (x^2+1)^7 \) with \( u^7 \).

Step 4: Integrate with respect to u

Now integrate \( 8 \int u^7 \, du \). Use the power rule for integration: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). Applying this, we get \( 8 \times \frac{u^8}{8} + C = u^8 + C \).

Step 5: Substitute Back to Original Variable

Replace \( u \) with \( x^2 + 1 \) to express the integral in terms of \( x \): \( (x^2 + 1)^8 + C \).

Step 6: Final Answer

The indefinite integral \( \int 16x(x^2+1)^7 \, dx \) is \( (x^2+1)^8 + C \).

Key Concepts

Method of Substitution

The method of substitution is a powerful technique in calculus, which helps to simplify the process of finding indefinite integrals, especially when dealing with composite functions. At its core, the substitution method involves changing variables to make integration more manageable. This is akin to solving a maze by transforming it into a straight line.

When you encounter an integral that looks complicated, such as \( \int 16x(x^2+1)^7 \, dx \), start by identifying parts of the integral that might fit together as a single expression. Here, the expression \( x^2 + 1 \) appears repeatedly, so we choose it for substitution. We set \( u = x^2 + 1 \), and this simplifies our work by reducing the complex integral into an easier form.

• Pick a substitution that simplifies the integral.
• Find the derivative \( du \) to replace \( dx \).
• Express the entire integral in terms of \( u \) and \( du \).

The next step is crucial. By rewiring the integral in terms of \( u \), you often change a difficult integration problem into something more approachable.

Integration by Substitution

Integration by substitution joins the realms of integration and differentiation elegantly. It's particularly effective when you deal with composite functions, where one function is inside another. This technique can simplify the integral by transforming it into a basic form.

For the integral \( \int 16x(x^2+1)^7 \, dx \), after setting \( u = x^2 + 1 \), you find that \( du = 2x \, dx \). You notice the original integral needs \( 16x \, dx \), but \( du \) only gives you \( 2x \, dx \). So, a small rearrangement is required: multiply \( du \) by 8 to match \( 16x \, dx \). Thus, \( 16x \, dx = 8 \, du \).

• The resulting integral of simple form becomes \( \int 8u^7 \, du \).
• Use basic integration rules, like the power rule, to solve.

This method makes integration doable by expressing the integral in simpler terms. Once integrated, remember to translate back into the original variable, \( x \), for your final answer, which in this case returns to \( (x^2+1)^8 + C \). This step ensures you're linking your solution back to the initial question.

Composite Function

A composite function, in essence, is a combination of two functions where one function is applied inside another. Think of it as a function within a function. These functions are especially common in calculus and can appear daunting at first, but with the right techniques, they become much easier to handle.

In the exercise, \( x^2 + 1 \) represents the inner function embedded within the larger power, creating another function. This makes \( (x^2 + 1)^7 \) a composite function. Resolving such a function requires identifying the inner and outer layers, which is essential for substitution.

• Identify the inner "core" function you can differentiate: \( u = x^2 + 1 \).
• Place this function as your substitution utility.
• Replace occurrences in the integral to rewrite in simpler terms.

Understanding and manipulating composite functions opens up the path for easier integration. Once you've mastered identifying where one function lives inside another, tackling these mathematical problems becomes much simpler.