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Chapter 7: Problem 42

Compute the indefinite integrals \(41-52\). $$ \int \tan 5 x d x $$

Short Answer

Expert verified
\( -\frac{1}{5} \ln |\cos(5x)| + C \)

Step by step solution

01

Recall the Indefinite Integral Formula for Tangent

To integrate \( \tan kx \), we can use the formula \( \int \tan(kx)\,dx = -\frac{1}{k} \ln |\cos(kx)| + C \), where \( C \) is the constant of integration. Here, \( k = 5 \).
02

Apply the Formula

Substitute \( k = 5 \) into the formula. So, \( \int \tan(5x)\,dx = -\frac{1}{5} \ln |\cos(5x)| + C \).
03

Write the Solution

The indefinite integral of \( \tan 5x \) is \( -\frac{1}{5} \ln |\cos(5x)| + C \).

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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 methods used to transform complex integrals into simpler ones. This makes finding the integral of a function easier. Some of the important techniques include:
  • Substitution: Simplifies the integrand by changing variables, turning complex functions into simpler forms.
  • Integration by Parts: Uses the product rule for differentiation in reverse.
  • Partial Fractions: Breaks down complex rational functions into simpler partial fractions.
In the exercise here, we used a specific formula for the integral of a tangent function. This is a great time-efficient technique for dealing with trigonometric integrals like the one we solved from the textbook.
Trigonometric Integrals
Trigonometric integrals are a type of integral where trigonometric functions form part of the integrand. They often appear in problems related to waveforms, oscillations, and circular motion. To solve them, you might use special identities or direct integral formulas. For example:
  • sine and cosine integrals
  • tangent integrals
  • secant and cosecant integrals
The exercise uses the formula for the integral of the tangent function, which is derived from the identity relating tangent and cosine: \[ \int \tan(kx)\,dx = -\frac{1}{k} \ln |\cos(kx)| + C \]Trigonometric integrals can be tricky unless you know the proper identities and relationships between the functions.
Constant of Integration
The constant of integration, denoted as \( C \), plays a crucial role in indefinite integrals. It represents an arbitrary constant added to the antiderivative. In essence, since differentiation of a constant yields zero, adding \( C \) ensures the solution accounts for all possible original functions. Consider:
  • Without \( C \): The solution represents only one of the infinite possible functions.
  • With \( C \): The solution captures the entire family of functions.
This is crucial in applications involving boundary conditions or initial value problems. The undefined nature of \( C \) emphasizes the infinite possibilities of antiderivates for any given function. In our exercise, \( C \) ensures that every function of the form \(-\frac{1}{5} \ln |\cos(5x)| + C \) is covered by the integral solution.

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Most popular questions from this chapter

Decide convergence or divergence. Compute the integrals that converge. $$ \int_{0}^{1} \frac{d x}{x^{\pi}} $$

Integrate by substitution. Change \(\theta\) back to \(x\). $$ \int \frac{d x}{x^{2} \sqrt{x^{2}+1}} $$

In \(35-40\) derive "reduction formulas" from higher to lower powers. $$ \int(\ln x)^{n} d x=x(\ln x)^{n}-n \int(\ln x)^{n-1} d x $$

Express the rational functions \(3-16\) as partial fractions: $$ \frac{1}{(x-1)\left(x^{2}+1\right)} $$

Express the rational functions \(3-16\) as partial fractions: $$ \frac{1}{x^{3}-x} $$
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