Question

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.) \(\int \tan (5 x) d x\)

Short answer

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

Step-by-step solution

Understand the Integral

We need to evaluate the integral \( \int \tan(5x) \, dx \). The integrand is the tangent function of a linear expression \( 5x \). Recall that the function \( \tan(x) \) can be expressed as \( \frac{\sin(x)}{\cos(x)} \).

Use a Substitution

To integrate \( \tan(5x) \), we use the substitution method. Let \( u = 5x \), so that \( du = 5 \, dx \) or \( dx = \frac{1}{5} du \). Rewrite the integral in terms of \( u \):\[\int \tan(u) \cdot \frac{1}{5} \, du = \frac{1}{5} \int \tan(u) \, du\]

Integrate the Tangent Function

The integral of \( \tan(u) \) is known to be \( -\ln|\cos(u)| + C \) based on the derivative relationships for logarithmic functions. Thus, we have:\[\frac{1}{5} \int \tan(u) \, du = \frac{1}{5} [-\ln|\cos(u)|] + C\]

Substitute Back the Original Variable

Replace \( u \) back with \( 5x \) in the expression:\[\int \tan(5x) \, dx = -\frac{1}{5} \ln|\cos(5x)| + C\]

Verification with CAS

Cross-check the solution using a Computer Algebra System (CAS) to ensure accuracy. The CAS should confirm:\[-\frac{1}{5} \ln|\cos(5x)| + C\]

Key concepts

Integration Techniques

Integration techniques are essential tools for solving complex integrals. When we encounter trigonometric functions in integrals, special methods are used. For example, an integral involving powers of trigonometric functions often requires strategic manipulation for simplification.

These techniques include:

• Substitution: Changing variables to simplify the integration process. It works by turning the integrand into something more manageable.
• Parts Integration: A method that is particularly useful when the integrand is a product of two functions.
• Trig Identities: Using identities to rewrite trigonometric expressions into a form that is easier to integrate.

By applying these techniques efficiently, you can handle a wide variety of integrals in a systematic and effective manner.

Substitution Method

The substitution method is a powerful and commonly used technique in calculus for evaluating integrals. This method works particularly well when the integrand includes a function and its derivative.

In this method:

• You select a part of the integrand to substitute. Let's say "changing the variable," typically denoted as \( u \).
• By defining \( u \), you find \( du \) to replace another part of the integrand.
• This substitution transforms the integral into a simpler form, which can be easily integrated.
• After integration, don't forget to substitute the original variable back into the final expression.

For example, in the integral \( \int \tan(5x) \, dx \), we set \( u = 5x \). Hence, \( dx \) becomes \( \frac{1}{5} du \), and the integral simplifies to a friendlier form \( \frac{1}{5} \int \tan(u) \, du \). This makes the integration process straightforward and less error-prone.

Tangent Function Integration

Integrating the tangent function can be tricky but is manageable with the right approach. The tangent function \( \tan(x) \) can be expressed in terms of sine and cosine: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).

For the integral \( \int \tan(x) \, dx \), instead of leaving it in its original form, use the identity and substitution methods to simplify it. Once expressed as a logarithmic function, the integral becomes: \[-\ln|\cos(x)| + C\]This comes from recognizing that the derivative of \( -\ln|\cos(x)| \) is \( \tan(x) \).

This means:

• Recognize that you need to deal with \( \tan(x) \) by using its identity.
• Once simplified, integrate the expression resulting in a logarithmic function.

Through practice, this process becomes a familiar and useful tool for managing integrals involving the tangent function.

Computer Algebra Systems (CAS)

Computer Algebra Systems (CAS) are highly advanced tools that perform symbolic mathematics. They can execute complex computations quickly and accurately, making them invaluable in checking manual calculations.

With the aid of CAS:

• You can verify solutions to integrals, ensuring whether the manual work is correct.
• These systems handle large and complicated algebraic expressions that might be cumbersome for manual methods.
• Common examples include software tools like Mathematica, Wolfram Alpha, and MATLAB.

Using CAS can provide peace of mind when tackling difficult integrals, as these systems confirm your solutions swiftly. By comparing CAS results with manual calculations, students enhance both their problem-solving confidence and comprehension.