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Chapter 8: Problem 39

Find the integral involving secant and tangent. \(\int \sec ^{3} x \tan x d x\)

Short Answer

Expert verified
The solution to the integral \(\int \sec ^{3} x \tan x d x\) is \(\frac{1}{4}\sec^{4}x + C\).

Step by step solution

01

Choose the appropriate substitution

In this case, one can choose \(u = \sec x\). This simplifies the integral, as it allows us to substitute all instances of \(\sec x\) with \(u\). Further, we need to find the derivative of \(u = \sec x\), which is \(du = \sec x \tan x dx\).
02

Perform the substitution

Using our chosen substitution, the original integral simplifies to \(\int u^{3} du\), which is a much simpler form that we can solve.
03

Solve the simplified integral

The integral \(\int u^{3} du\) can now be easily solved using the power rule for integration, which states that \(\int x^{n} dx = \frac{1}{n+1} x^{n+1} + C\), where \(n\) is any real number and \(C\) is the constant of integration. Applying this to our simplified integral, we get \(\frac{1}{4} u^{4} + C\).
04

Substitute the original variable back

Finally, we substitute the original variable back into our solved integral. This gives us \(\frac{1}{4}\sec^{4}x + C\), which is the solution to the integral.

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Key Concepts

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

Integration Techniques
When learning to integrate functions, a variety of techniques is available. Each technique suits different types of functions. By mastering these techniques, you become a versatile problem solver.

One common technique is the **power rule for integration**. This is highly effective when dealing with polynomials. The formula is \[ \int x^n \, dx = \frac{1}{n+1}x^{n+1} + C \]where \( n \) is a real number and \( C \) is the constant of integration.

Another method involves choosing the right **substitution**. This transforms an integral into a simpler form, making it more manageable to solve.
  • Always look for ways to apply substitutions to simplify complex integrals.
  • Trigonometric integrals often benefit from this technique.
With practice, recognizing which technique to apply becomes second nature.
Trigonometric Integrals
Integral calculus frequently encounters trigonometric functions, requiring particular strategies. Trigonometric integrals, especially, may seem daunting at first.

The functions secant and tangent often appear together. They have a special derivative relationship: \[\frac{d}{dx}(\sec x) = \sec x \tan x.\]This relationship offers a direct path for substitution during integration.

Here are some general tips:
  • Look for products of secant and tangent to exploit their derivative relationships.
  • Rewrite functions in terms of a single trigonometric function if possible.
Understanding these relationships forms the basis of solving trigonometric integrals effectively.
Substitution Method
The substitution method is an essential part of integral calculus, providing a powerful tool for simplifying difficult integrals. The general idea is to replace a part of the integral with a single variable, easing computation.

For example, in the exercise:\( \int \sec^3 x \tan x \, dx \), set \( u = \sec x \), so \( du = \sec x \tan x \, dx \). The integral becomes \( \int u^3 \, du \), which is straightforward with the power rule.

Fundamental steps for substitution:
  • Choose a substitution that reduces the integral to a simpler form.
  • Express \( dx \) in terms of \( du \).
  • Perform the integration.
  • Convert back to the original variable.
By practicing substitution, you enhance your ability to tackle a wide range of integrals.

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Most popular questions from this chapter

Find the value of \(c\) that makes the function continuous at \(x=0\). $$ f(x)=\left\\{\begin{array}{ll} \frac{4 x-2 \sin 2 x}{2 x^{3}}, & x \neq 0 \\ c, & x=0 \end{array}\right. $$

Find the capitalized cost \(C\) of an asset (a) for \(n=5\) years, (b) for \(n=10\) years, and (c) forever. The capitalized cost is given by $$ C=C_{0}+\int_{0}^{n} c(t) e^{-r t} d t $$ where \(C_{0}\) is the original investment, \(t\) is the time in years, \(r\) is the annual interest rate compounded continuously, and \(c(t)\) is the annual cost of maintenance. $$ \begin{aligned} &C_{0}=\$ 650,000 \\ &c(t)=\$ 25,000(1+0.08 t) \\ &r=0.06 \end{aligned} $$

Work A hydraulic cylinder on an industrial machine pushes a steel block a distance of \(x\) feet \((0 \leq x \leq 5)\), where the variable force required is \(F(x)=2000 x e^{-x}\) pounds. Find the work done in pushing the block the full 5 feet through the machine.

True or False? , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is continuous on \([0, \infty)\) and \(\int_{0}^{\infty} f(x) d x\) diverges, then \(\lim _{x \rightarrow \infty} f(x) \neq 0\).

Apply the Extended Mean Value Theorem to the functions \(f\) and \(g\) on the given interval. Find all values \(c\) in the interval \((a, b)\) such that $$\frac{f^{\prime}(c)}{g^{\prime}(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}$$ Functions \(\quad\) Interval $$ f(x)=\frac{1}{x}, \quad g(x)=x^{2}-4 $$ $$ [1,2] $$
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