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Chapter 9: Problem 16

Determine the integrals by making appropriate substitutions. \(\int \frac{x}{\sqrt{x^{2}+1}} d x\)

Short Answer

Expert verified
\[ \int \frac{x}{\sqrt{x^2 + 1}} dx = \sqrt{x^2 + 1} + C \]

Step by step solution

01

Identify the Substitution

Recognize the form of the integrand to decide on a substitution. In this case, let鈥檚 use the substitution method where we set a new variable for simplifying the integrand. Let us set \[ u = x^2 + 1 \]
02

Compute the Derivative

Take the derivative of the substitution variable with respect to x. This will help us replace the dx term in the integral. So we get: \[ du = 2x \, dx \] or \[ dx = \frac{du}{2x} \]
03

Substitute in the Integral

Replace the terms in the original integral with the new variable u. We get: \[ \int \frac{x}{\sqrt{x^2+1}} dx = \int \frac{x}{\sqrt{u}} \cdot \frac{du}{2x} \] The x terms cancel out, leaving us with: \[ \frac{1}{2} \int \frac{1}{\sqrt{u}} du \]
04

Simplify the Integral

Simplify the integral before solving it. The integral now becomes: \[ \frac{1}{2} \int u^{-1/2} du \]
05

Integrate

Integrate the simplified expression. Use the power rule for integration: \[ \int u^n du = \frac{u^{n+1}}{n+1} + C \] Therefore, \[ \frac{1}{2} \int u^{-1/2} du = \frac{1}{2} \cdot 2u^{1/2} + C = \sqrt{u} + C \]
06

Substitute Back

Substitute back the original variable x into the solution: \[ \sqrt{u} + C = \sqrt{x^2 + 1} + C \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integrals
Integrals are a fundamental concept in calculus. They are used to find the area under a curve, among other things.
When we talk about integrals, we often mean solving an expression that involves a function and a 'dx' term. Here's a breakdown:
  • The integral symbol \(\int\) tells us that we are looking to find the area under a curve.
  • The expression inside the integral, such as \(\frac{x}{\sqrt{x^{2}+1}} dx\), is the integrand.
  • The 'dx' indicates that we are integrating with respect to the variable x.
There are multiple techniques for solving integrals and understanding them is key to mastering calculus.
Substitution Method
The substitution method is a powerful tool in calculus for simplifying integrals. It works by changing the variable of integration to make the integral easier to solve. Here鈥檚 how it works:
  • Identify the substitution: We look at the integrand and decide on a suitable substitution. In our example, setting \(\u = x^2 + 1\) simplifies our integrand.
  • Compute the derivative: We take the derivative of u with respect to x to find du. For instance, \(\du = 2x dx\).
  • Replace variables in the integral: We substitute u and du back into the integral to create a simpler form. This might look like \(\frac{1}{2} \int u^{-1/2} du\).
  • Solve the new integral: The simplified integral is now often easier to integrate, using standard rules like the power rule.
  • Substitute back the original variable: After integrating, we replace u back with x. So, \(\sqrt{u} + C\) would turn back into \(\sqrt{x^2 + 1} + C\).
By following these steps, the integral becomes much easier to handle.
Calculus
Calculus is the branch of mathematics that deals with rates of change and the accumulation of quantities. It's divided into two main parts: Differential Calculus and Integral Calculus.
  • **Differential Calculus:** Focuses on the concept of the derivative, which represents the rate of change of a function.
  • **Integral Calculus:** Deals with integrals and their properties. It's all about finding the total amount accumulated, such as the area under a curve.
Most real-world changes we see around us can be modeled using calculus. This includes everything from the speed of a car to the growth rates of populations. Understanding calculus can open up a deeper understanding of the mechanics of the world and various disciplines such as physics, engineering, economics, and beyond.
Calculus uses various techniques to solve problems, and the substitution method for integrals is just one of many.

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Most popular questions from this chapter

Evaluate the following integrals using techniques studied thus far. \(\int\left(x e^{2 x}+x^{2}\right) d x\)

Find an approximate value by Simpson's rule. Express your answers to five decimal places. \(\int_{0}^{1} \frac{1}{x^{3}+1} d x ; n=2\)

Use substitutions and the fact that a circle of radius \(r\) has area \(\pi r^{2}+\) evaluate the following integrals. \(\int_{-\pi / 2}^{\pi / 2} \sqrt{1-\sin ^{2} x} \cos x d x\)

A continuous stream of income is produced at the rate of \(20 e^{1-.03 t}\) thousand dollars per year at. time \(t\), and invested money earns \(6 \%\) interest. (a) Write a definite integral that gives the present value of this stream of income over the time from \(t=2\) to \(t=5\) years. (b) Compute the present value described in part (a).

A volcano erupts and spreads lava in all directions. The density of the deposits at a distance \(t\) kilometers from the center is \(D(t)\) thousand tons per square kilometer, where \(D(t)=11\left(t^{2}+10\right)^{-2}\). Find the tonnage of lava deposited between the distances of 1 and 10 kilometers from the center.
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