By the integration by parts, we haveZ π−πf (x) cos(nx)dx = 1n f (x) sin(nx)π−π− 1nZ π−πf (x) sin(nx)dx

To solve the given integral using integration by parts, we will follow these steps:

Step 1: Identify the function to be integrated, which is f(x).

Step 2: Identify the function to be differentiated, which is cos(nx).

Step 3: Apply the integration by parts formula, which states that the integral of the product of two functions, u and v, is equal to the product of u and the integral of v, minus the integral of the derivative of u multiplied by the integral of v.

Step 4: Apply the integration by parts formula to the given integral:

∫[π, -π] f(x) cos(nx) dx = 1/n f(x) sin(nx) [π, -π] - (1/n) ∫[π, -π] f'(x) sin(nx) dx

Step 5: Simplify the integral:

∫[π, -π] f(x) cos(nx) dx = (1/n) f(x) sin(nx) [π, -π] - (1/n) ∫[π, -π] f'(x) sin(nx) dx

Step 6: Evaluate the definite integral by substituting the limits of integration:

∫[π, -π] f(x) cos(nx) dx = (1/n) f(x) sin(nx) |[π, -π] - (1/n) ∫[π, -π] f'(x) sin(nx) dx

Step 7: Simplify further if necessary.

Please note that the above steps are a general guide for solving integrals using integration by parts. The specific solution may vary depending on the given function f(x) and the limits of integration.