Derive the integral of cos^n (x) dx= (1/n)cos^(n-1) (x) sin(x)+((n-1)/n)(integral of cos^(n-2) (x) dx

To derive the integral of cos^n(x), we can use the method of integration by parts and reduction formula. Here's how you can do it step by step:

Step 1: Start with the given integral ∫cos^n(x) dx.

Step 2: Use integration by parts. Let u = cos^(n-1)(x) and dv = cos(x) dx. Then, differentiate u to get du = (n-1)cos^(n-2)(x)(-sin(x)) dx and integrate dv to get v = sin(x). Applying the integration by parts formula: ∫u dv = uv - ∫v du.

Step 3: Apply the integration by parts formula to the given integral:

∫cos^n(x) dx = ∫u dv = uv - ∫v du
= cos^(n-1)(x)sin(x) - ∫sin(x)(n-1)cos^(n-2)(x)(-sin(x)) dx
= cos^(n-1)(x)sin(x) + (n-1)∫sin^2(x)cos^(n-2)(x) dx

Step 4: Convert sin^2(x) to 1 - cos^2(x) using the Pythagorean identity:

∫cos^n(x) dx = cos^(n-1)(x)sin(x) + (n-1)∫(1 - cos^2(x))cos^(n-2)(x) dx

Step 5: Simplify and factor out cos^(n-1)(x):

∫cos^n(x) dx = cos^(n-1)(x)sin(x) + (n-1)∫cos^(n-2)(x) dx - (n-1)∫cos^n(x) dx

Step 6: Rearrange the equation to isolate the integral of cos^n(x):

∫cos^n(x) dx + (n-1)∫cos^n(x) dx = cos^(n-1)(x)sin(x) + (n-1)∫cos^(n-2)(x) dx

Step 7: Combine the integrals on the left-hand side:

n∫cos^n(x) dx = cos^(n-1)(x)sin(x) + (n-1)∫cos^(n-2)(x) dx

Step 8: Divide both sides by n to solve for the integral of cos^n(x):

∫cos^n(x) dx = (1/n)cos^(n-1)(x)sin(x) + ((n-1)/n)∫cos^(n-2)(x) dx

And that's the derived formula for the integral of cos^n(x).