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Chapter 9: Problem 7

Evaluate the following integrals: \(\int \frac{x}{\sqrt{x+1}} d x\)

Short Answer

Expert verified
\[ \frac{2}{3} (x+1)^{3/2} - 2 (x+1)^{1/2} + C \]

Step by step solution

01

Substitution

Choose a substitution to simplify the integral. Let鈥檚 set \( u = x + 1 \). Then, differentiate both sides with respect to \( x \): \( du = dx \).Also, notice that \( x = u - 1 \).
02

Rewrite the Integral

Substitute the expressions for \( x \) and \( dx \) in terms of \( u \) into the integral: \[ \int \frac{x}{\sqrt{x+1}} dx = \int \frac{u-1}{\sqrt{u}} du \].
03

Simplify the Integral

Separate the integral: \[ \int \frac{u-1}{\sqrt{u}} du = \int \frac{u}{\sqrt{u}} du - \int \frac{1}{\sqrt{u}} du \].Simplify the fractions: \[ \int u^{1/2} du - \int u^{-1/2} du \].
04

Integrate

Now integrate each term separately:\[ \int u^{1/2} du = \frac{2}{3} u^{3/2} \]\[ \int u^{-1/2} du = 2 u^{1/2} \].
05

Combine Results and Substitute Back

Combine the results of the integrals and substitute back \( u = x + 1 \):\[ \frac{2}{3} u^{3/2} - 2 u^{1/2} = \frac{2}{3} (x+1)^{3/2} - 2 (x+1)^{1/2} + C \].

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Key Concepts

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

substitution method
The substitution method, also known as *u-substitution*, is a technique used to simplify complex integrals by introducing a new variable. In our exercise, we set \( u = x + 1 \), which transforms the original integral into a more manageable form. This method works best when the integrand includes a composite function. After substituting, we replace all instances of the original variable and its differentials with the new variable and its differentials.
  • Step 1: Identify a substitution that simplifies the integral. In this case, it's \( u = x + 1 \).
  • Step 2: Differentiate to find \( du = dx \).
  • Step 3: Substitute back into the integral.
Once you solve the integral in terms of \( u \), don't forget to substitute back the original variable to complete the process.
definite integrals
Definite integrals are used to find the exact area under a curve between two points. They are expressed with upper and lower limits. If we were solving a definite integral for the given exercise, we would have limits on our integral.
Some key points about definite integrals include:
  • They provide a numerical value representing the area.
  • The limits of integration are crucial as they define the scope of the area.
  • After performing the integration, always apply the limits to find the final result.
Definite integrals differ from indefinite integrals, which do not have fixed limits and include an arbitrary constant, represented as 'C'.
indefinite integrals
Indefinite integrals represent a family of functions and include an arbitrary constant *C*. This constant arises because the process of differentiation of a constant is zero, making it impossible to determine its value from the integral alone.
Indefinite integrals in this context:
  • The exercise involves finding the indefinite integral of \( \frac{x}{ \sqrt{x+1}} \ dx \).
  • Using the substitution method, we simplify and integrate in terms of the new variable 'u'.
  • After integrating, we re-substitute to return to the original variable.
  • The result includes a '+ C', symbolizing the family of possible functions.
This constant is essential when interpreting the integral as part of initial value problems and general solutions.
integration by parts
Integration by parts is a valuable technique used when integrating the product of two functions. It is based on the integration analogue of the product rule for differentiation and is expressed as: \[ \int u dv = uv - \int v du \]
Steps involved in integration by parts often include:
  • Selecting the parts of the integrand as \( u \) and \( dv \).
  • Differentiating \( u \) to find \( du \), and integrating \( dv \) to obtain \( v \).
  • Substituting into the integration by parts formula.
While it isn't directly used in our specific exercise involving substitution, it's important to understand this method as it often complements substitution in more complex integrations. This technique helps solve integrals where substitution alone is not sufficient.

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Most popular questions from this chapter

Suppose that the population density function for a city is \(40 e^{-.5 t}\) thousand people per square mile. Let \(P(t)\) be the total population that lives within \(t\) miles of the city center, and let \(\Delta t\) be a small positive number. (a) Consider the ring about the city whose inner circle is at \(t\) miles and outer circle is at \(t+\Delta t\) miles. The text shows that the area of this ring is approximately \(2 \pi t \Delta t\) square miles. Approximately how many people live within this ring? (Your answer will involve \(t\) and \(\Delta t .\).) (b) What does $$\frac{P(t+\Delta t)-P(t)}{\Delta t}$$ approach as \(\Delta t\) tends to zero? (c) What does the quantity \(P(5+\Delta t)-P(5)\) represent? (d) Use parts (a) and (c) to find a formula for $$\frac{P(t+\Delta t)-P(t)}{\Delta t}$$ and from that obtain an approximate formula for the derivative \(P^{\prime}(t) .\) This formula gives the rate of change of total population with respect to the distance \(t\) from the city center. (c) Given two positive numbers \(a\) and \(b\), find a formula involving a definite integral, for the number of people who live in the city between \(a\) miles and \(b\) miles of the city center. [Hint: Use part (d) and the fundamental theorem of calculus to compute \(P(b)-P(a) .]\)

Approximate the following integrals by the midpoint rule, the frapezoidal rule, and Simpson's rule. Then, find the exact value by integration. Express your answers to five decimal places. \(\int_{0}^{2} 2 x e^{x^{2}} d x ; n=4\)

Consider \(\int_{1}^{2} f(x) d x\), where \(f(x)=3 \ln x\). (a) Make a rough sketch of the graph of the fourth derivative of \(f(x)\) for \(1 \leq x \leq 2\). (b) Find a number \(A\) such that \(\left|f^{\prime \prime \prime \prime}(x)\right| \leq A\) for all \(x\) satisfying \(1 \leq x \leq 2\). (c) Obtain a bound on the error of using Simpson's rule with \(n=2\) to approximate the definite integral. (d) The exact value of the definite integral (to four decimal places) is \(1.1589\), and Simpson's rule with \(n=2\) gives \(1.1588\). What is the error for the approximation by Simpson's rule? Does this error satisfy the bound obtained in part (c)? (c) Redo part (c) with the number of intervals tripled to \(n=6\). Is the bound on the error divided by three?

Find an approximate value by Simpson's rule. Express your answers to five decimal places. \(\int_{0}^{2} \sqrt{\sin x} d x ; n=5\)

Evaluate the following improper integrals whenever they are convergent. \(\int_{3}^{\infty} \frac{x^{2}}{\sqrt{x^{3}-1}} d x\)
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