integrate 1/(3sin(x) - 4cos(x)) dx

integrate 1/(3sin(x) - 4cos(x)) dx We are asked to evaluate the integral: ∫ 1 / (3sin(x) - 4cos(x)) dx Step 1: The integral involves a combination of trigonometric functions, so we need to use substitution to simplify it. Step 2: Let’s try the substitution method. Let’s set: u = 3sin(x) - 4cos(x) Step 3: Now, differentiate u with respect to x: du/dx = 3cos(x) + 4sin(x) So, we get: du = (3cos(x) + 4sin(x)) dx Step 4: We want the numerator to match du, but it currently has 1 in the numerator and only \( 3\sin(x) - 4\cos(x) \) in the denominator. This suggests that we can try to match terms using a standard method of integrating trigonometric functions. However, solving this exactly without a specific setup can become complex. We can express the denominator in a more convenient form, possibly using trigonometric identities, or attempt more advanced methods such as the method of multiplying by a suitable expression to facilitate simplification. Alternatively, this form is commonly solved using the following general solution for integrals of the type \( \int \frac{dx}{a\sin(x) + b\cos(x)} \): ∫ 1 / (a sin(x) + b cos(x)) dx = (1/√(a² + b²)) * ln | a sin(x) + b cos(x) + √(a² + b²) | + C Step 5: In our case, a = 3 and b = -4, so: ∫ 1 / (3sin(x) - 4cos(x)) dx = (1 / √(3² + (-4)²)) * ln | 3sin(x) - 4cos(x) + √(3² + (-4)²) | + C Step 6: Simplify the constants: ∫ 1 / (3sin(x) - 4cos(x)) dx = (1 / 5) * ln | 3sin(x) - 4cos(x) + 5 | + C Thus, the integral is: ∫ 1 / (3sin(x) - 4cos(x)) dx = (1 / 5) * ln | 3sin(x) - 4cos(x) + 5 | + C