Question

Find the indefinite integral. (Use C for the constant of integration.) ∫8/cos(θ} - 1 × dθ _________

Answer 1

The indefinite integral of 8/cos(θ) - 1 with respect to θ is 8 ln|sec(θ) + tan(θ)| - θ + C, where C represents the constant of integration.

Step-by-step explanation:

To find the indefinite integral of the function 8/cos(θ) - 1, you would integrate it with respect to θ. The integral of 8/cos(θ) is 8 sec(θ), where sec(θ) is the secant function, which is the reciprocal of the cosine function. The integral of -1 is simply -θ. Therefore, the indefinite integral of the given function is:

∫ (8/cos(θ) - 1) dθ = 8∫ sec(θ) dθ - ∫ dθ = 8 ln|sec(θ) + tan(θ)| - θ + C

Here, we used the integral identity that the antiderivative of sec(θ) is ln|sec(θ) + tan(θ)|, and we noted that the antiderivative of a constant is just the constant times the variable of integration. Don't forget to include the constant of integration represented by C when expressing the final answer for an indefinite integral.