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Chapter 8: Problem 35

Evaluate the following integrals. $$\int \sec ^{2} 3 x d x$$

Short Answer

Expert verified
\frac{1}{3} \tan(3x) + C

Step by step solution

01

Identify the Integral Form

Recognize that the integrand is in the form \(\text{sec}^2(kx)\). The antiderivative of \(\text{sec}^2(x)\) is \(\tan(x)\).
02

Use Substitution Method

Set \(u = 3x\). Then, differentiate to find \(du = 3dx\) or \(\frac{1}{3} du = dx\).
03

Rewrite the Integral

Substitute \(3x\) with \(u\) and \(dx\) with \(\frac{1}{3} du\). The integral becomes \(\frac{1}{3} \int \text{sec}^2(u) du\).
04

Integrate

Integrate \(\text{sec}^2(u)\) to get \(\tan(u)\). Thus, the integral becomes \(\frac{1}{3} \tan(u) + C\).
05

Substitute Back in Terms of x

Replace \(u\) with \(3x\) to get the final answer \(\frac{1}{3} \tan(3x) + C\).

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Key Concepts

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

Integration Techniques
Integration techniques are diverse strategies used to find the integral of a function. Techniques involve simplifications, substitutions, and recognizing patterns. Consider the integral \(\bigint \text{sec}^2 (3x) \text{d} x\). We identify the integral form and apply an appropriate technique. Here, the antiderivative for \(\text{sec}^2(x)\) is \(\tan(x)\). Finding antiderivatives directly is impractical for complex forms, so we use techniques like:
  • Substitution
  • Integration by parts
  • Partial fractions
Identifying patterns and using specific strategies makes integration manageable and efficient.
Substitution Method
The substitution method simplifies integrals by substituting variables, transforming the integral into a simpler form. This is useful when the integrand matches the derivative of another function. Let's illustrate this using \(\int \text{sec}^2 (3x) \text{d} x\). Here鈥檚 how substitution works step-by-step:
  • Set \(u = 3x\)
  • Differentiate: \(\text{d}u = 3\text{d}x\)
  • Solve for dx: \(\text{d}x = \frac{1}{3} \text{d}u\)
Now substitute: \(\int \text{sec}^2 (u) \frac{1}{3} \text{d}u\). This reduces the problem to a simpler integral, \(\frac{1}{3} \int \text{sec}^2 (u) \text{d}u\). Finally, integrate and substitute back to the original variable.
Trigonometric Integrals
Trigonometric integrals involve integrands with trigonometric functions like sine, cosine, secant, and tangent. Understanding their antiderivatives and identities is essential. For example, the integral \(\bigint \text{sec}^2 (3x) \text{d} x\), relates to the fact that \(\text{sec}^2(x)\) is the derivative of \(\tan(x)\). Recognizing these relationships is key to solving such integrals efficiently. Trigonometric identities help simplify expressions and reveal integrable forms. Remember these tips:
  • Know basic trigonometric antiderivatives: \(\int \text{sec}^2(x) \text{d}x = \tan(x) + C\)
  • Use identities to transform integrals
  • Apply substitution for complex arguments
These strategies will make handling trigonometric integrals straightforward.

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Most popular questions from this chapter

Differentiate (with respect to \(t\) or \(x\) ): $$y=\ln (\cos t)$$

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As \(h\) approaches \(0,\) what value is approached by $$\frac{\cos (\pi+h)+1}{h} ?$$ \([\text {Hint:} \cos \pi=-1]\)

Find the equation of the line tangent to the graph of \(y=3 \sin 2 x-\cos 2 x\) at \(x=3 \pi / 4.\)

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