Question

Find the indefinite integral cosx / 1 + 4sinx dx

a) − 1 / 4 ln ∣1+4sin(x)∣ + C
b) 1 / 4 ln ∣1+4sin(x)∣ + C
c) − 1 / 4 ln ∣1-4sin(x)∣ + C
d) 1 / 4 ln ∣1-4sin(x)∣ + C

Answer 1

The indefinite integral of cos(x) / (1 + 4sin(x)) is solved by substitution, leading to the answer 1/4 ln|1 + 4sin(x)| + C, which is option b.

Step-by-step explanation:

The question asks to find the indefinite integral of the function cos(x) / (1 + 4sin(x)). To solve this, we can use a substitution method. Let u = 1 + 4sin(x), then du = 4cos(x)dx. Now, our integral can be rewritten as 1/4 ∫ du/u, which is a basic form of the natural logarithm integral. Performing the integration, we get 1/4 ln|u| + C.

Reverting back to the original variable gives us 1/4 ln|1 + 4sin(x)| + C. This corresponds to option b, which is the correct answer.