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Chapter 7: Problem 3

Evaluate the integrals by making appropriate \(u\) -substitutions and applying the formulas reviewed in this section. $$\int x \sec ^{2}\left(x^{2}\right) d x$$

Short Answer

Expert verified
\( \frac{1}{2} \tan(x^2) + C \)

Step by step solution

01

Identify the function for substitution

Look at the integrand \( x \sec^2(x^2) \). Notice that the inner function \( x^2 \) inside the \( \sec^2 \) is a good candidate for substitution since its derivative \( 2x \) appears in the integrand with the factor \( x \).
02

Perform the substitution

Set \( u = x^2 \). Then, differentiate both sides to get \( du = 2x \, dx \), or equivalently \( \frac{1}{2}du = x \, dx \). Substitute \( u \) and \( du \) into the integral.
03

Rewrite the integral in terms of \(u\)

Substitute \( u = x^2 \) and \( \frac{1}{2}du = x \, dx \) into the integral:\[ \int x \sec^2(x^2) \, dx = \int \sec^2(u) \cdot \frac{1}{2} du = \frac{1}{2} \int \sec^2(u) \, du \]
04

Integrate the new expression

Recall that the integral of \( \sec^2(u) \) is \( \tan(u) \). Thus, integrate to get:\[ \frac{1}{2} \int \sec^2(u) \, du = \frac{1}{2} \tan(u) + C \] where \( C \) is the constant of integration.
05

Substitute back to the original variable

Replace \( u \) with the original expression \( x^2 \):\[ \frac{1}{2} \tan(x^2) + C \]

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Key Concepts

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

u-substitution method
When we encounter complex integrals, the **u-substitution method** can be a powerful technique to simplify the process. Think of it as a way to transform a difficult integral into an easier one by changing variables, much like a change of perspective. In our example, we have the integral \( \int x \sec^2(x^2) \, dx \). The expression inside the trigonometric function \( \sec^2(x^2) \) suggests that substitution might make integration simpler.

Here's how to tackle it:
  • **Identify the inner function:** Look for the part of the integrand that is inside another function, often a composition such as \( x^2 \) in this case.
  • **Differentiate the inner function:** We find that the derivative of \( x^2 \) is \( 2x \), which is almost exactly what appears in the integral, apart from a constant factor.
  • **Substitute and integrate:** Set \( u = x^2 \), leading to \( du = 2x \, dx \) or \( x \, dx = \frac{1}{2} du \). This substitution transforms the integral into a more straightforward form: \( \frac{1}{2} \int \sec^2(u) \, du \).
By using u-substitution, integration can be performed more easily, helping to solve integrals that initially seem complex.
trigonometric integrals
**Trigonometric integrals** play a vital role in calculus, especially when dealing with periodic functions defined by sine, cosine, tangent, and their reciprocals. Our original problem involves the trigonometric function \( \sec^2(x^2) \), which is the derivative of another useful function.

Understanding this connection:
  • **Recognize relationships:** Knowing that the integral of \( \sec^2(u) \) is \( \tan(u) \) simplifies the process significantly, because this relationship comes from the fundamental derivatives and integrals list.
  • **Navigate transformations:** By changing the variable using u-substitution, it becomes evident that the trigonometric integral is manageable, leading directly to \( \frac{1}{2} \tan(u) + C \).
  • **Solve and revert:** After integrating this simpler trigonometric form, we substitute back to the original variable to solve the integral completely.
Utilizing known integrals helps in not only solving them faster but also understanding how these periodic functions behave under integration.
integration techniques
In calculus, mastering various **integration techniques** is essential. They provide tools for solving a range of integrals, from simple polynomials to more complex trigonometric or exponential expressions. Our example demonstrates the blend of these techniques:

Key points include:
  • **Selecting a method:** Choose the integration technique that simplifies the problem, such as u-substitution for the given exercise.
  • **Combining methods:** Often a problem requires more than one technique, such as handling a trigonometric integral with a substitution method.
  • **Applying constants of integration:** Remember to include \( C \), the constant of integration, as it represents an infinite number of antiderivatives.
Such techniques allow us to break down challenging integrals into smaller manageable ones, which can be solved using basic integration knowledge or standard integral formulas.

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Most popular questions from this chapter

(a).Make an appropriate \(u\) -substitution of the form \(u=x^{1 / 2}\) or \(u=(x+a)^{1 / n},\) and then evaluate the integral. (b).If you have a CAS, use it to evaluate the integral, and then confirm that the result is equivalent to the one that you found in part (a). $$\int \frac{\sqrt{x}}{x+1} d x$$

Find the volume of the solid generated when the region enclosed by \(y=\tan x, y=1,\) and \(x=0\) is revolved about the \(x\) -axis.

The average speed, \(\bar{v},\) of the molecules of an ideal gas is given by $$\bar{v}=\frac{4}{\sqrt{\pi}}\left(\frac{M}{2 R T}\right)^{3 / 2} \int_{0}^{+\infty} v^{3} e^{-M v^{2} /(2 R T)} d v$$ and the root-mean-square speed, \(v_{\mathrm{rms}},\) by $$v_{\mathrm{rms}}^{2}=\frac{4}{\sqrt{\pi}}\left(\frac{M}{2 R T}\right)^{3 / 2} \int_{0}^{+\infty} v^{4} e^{-M v^{2} /(2 R T)} d v$$ where \(v\) is the molecular speed, \(T\) is the gas temperature, \(M\) is the molecular weight of the gas, and \(R\) is the gas constant. (a) Use a CAS to show that $$\int_{0}^{+\infty} x^{3} e^{-a^{2} x^{2}} d x=\frac{1}{2 a^{4}}, \quad a>0$$ and use this result to show that \(\bar{v}=\sqrt{8 R T /(\pi M)}\) (b) Use a CAS to show that $$\int_{0}^{+\infty} x^{4} e^{-a^{2} x^{2}} d x=\frac{3 \sqrt{\pi}}{8 a^{5}}, \quad a>0$$ and use this result to show that \(v_{\mathrm{rms}}=\sqrt{3 R T / M}\).

In each part, try to evaluate the integral exactly with a CAS. If your result is not a simple numerical answer, then use the CAS to find a numerical approximation of the integral. (a) \(\int_{-\infty}^{+\infty} \frac{1}{x^{8}+x+1} d x\) (b) \(\int_{0}^{+\infty} \frac{1}{\sqrt{1+x^{3}}} d x\) (c) \(\int_{1}^{+\infty} \frac{\ln x}{e^{x}} d x\) (d) \(\int_{1}^{+\infty} \frac{\sin x}{x^{2}} d x\)

Suppose that during a period \(t_{0} \leq t \leq t_{1}\) years, a company has a continuous income stream at a rate of \(I(t)\) dollars per year at time \(t\) and that this income is invested at an annual rate of \(r \%,\) compounded continuously. The value (in dollars) of this income stream at the end of the time period \(t_{0} \leq t \leq t_{1},\) called the stream's future value, can be calculated using $$F V=\int_{t_{0}}^{t_{1}} I(t) e^{r\left(t_{1}-t\right)} d t$$ The present value (in dollars) of the income stream is given by $$P V=\int_{t_{0}}^{t_{1}} I(t) e^{-r\left(t-t_{0}\right)} d t$$ The present value is the amount that, if put in the bank at time \(t=t_{0}\) at \(r \%\) compounded continuously, with no additional deposits, would result in a balance of \(F V\) dollars at time \(t=t_{1}\) That is, $$F V=P V \times e^{r\left(t_{1}-t_{0}\right)}$$ In each exercise, (a) find the future value \(F V\) for the given income stream \(I(t)\) and interest rate \(r\) and time period \(t_{0} \leq t \leq t_{1}\) (b) find the present value \(P V\) of the income stream over the time period; and (c) verify that \(F V\) and \(P V\) satisfy the relationship given above. $$I(t)=2000 t+400 e^{-t} ; r=8 \% ; 0 \leq t \leq 25$$
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