evaluate the indefinite integral (1-6sinx)/cosxdx

We want to evaluate the indefinite integral:

\[
\int \frac{1 - 6\sin(x)}{\cos(x)} \, dx
\]

To solve this, we can break the integral into two simpler parts:

\[
\int \frac{1 - 6\sin(x)}{\cos(x)} \, dx = \int \frac{1}{\cos(x)} \, dx - 6 \int \frac{\sin(x)}{\cos(x)} \, dx
\]

Let's solve each part separately:

1. \(\int \frac{1}{\cos(x)} \, dx = \int \sec(x) \, dx\)
2. \(-6 \int \frac{\sin(x)}{\cos(x)} \, dx = -6 \int \tan(x) \, dx\)

I will evaluate these integrals now.# Define the variable and functions
x = sp.symbols('x')
expr = (1 - 6*sp.sin(x)) / sp.cos(x)

# Compute the indefinite integral
indefinite_integral = sp.integrate(expr, x)

indefinite_integral.simplify()import sympy as sp

# Redefine the variable and the expression
x = sp.symbols('x')
expr = (1 - 6 * sp.sin(x)) / sp.cos(x)

# Compute the indefinite integral
indefinite_integral = sp.integrate(expr, x)

# Simplify the result if possible
indefinite_integral.simplify()The indefinite integral of \( \frac{1 - 6\sin(x)}{\cos(x)} \, dx \) is:

\[
\int \frac{1 - 6\sin(x)}{\cos(x)} \, dx = -6\log\left(\frac{2}{\cos(x) + 1}\right) + 5\log\left(\tan\left(\frac{x}{2}\right) - 1\right) + 7\log\left(\tan\left(\frac{x}{2}\right) + 1\right) + C
\]

where \( C \) is the constant of integration.

If you have any questions or need further clarification, feel free to ask!

Here are 5 related questions:
1. How can you verify the result by differentiating it?
2. What are common techniques for integrating trigonometric expressions?
3. What substitution might simplify the original integral further?
4. How does the integral relate to inverse trigonometric functions?
5. Can this integral be expressed using other trigonometric identities?

**Tip:** When dealing with trigonometric integrals, try using substitutions like \( u = \tan(x/2) \) or known identities to simplify the process.

Learn how to evaluate the indefinite integral (1 - 6sin(x))/cos(x) step-by-step. This problem involves calculus concepts such as trigonometric integrals, secant function integration, and tangent function integration. Suitable for advanced high school students and calculus enthusiasts.

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