Use reduction formula to find the value of the following integrals:I. ∫ sec6 𝑥 𝑑𝑥𝜋40II. ∫ sin4𝑥 𝑐𝑜𝑠5𝑥 𝑑𝑥

To find the value of the given integrals using the reduction formula, we will follow these steps:

I. ∫ sec^6(x) dx

Step 1: Recall the reduction formula for ∫ sec^n(x) dx: ∫ sec^n(x) dx = (1/(n-1)) * sec^(n-2)(x) * tan(x) + (n-2)/(n-1) * ∫ sec^(n-2)(x) dx

Step 2: Apply the reduction formula to the given integral: ∫ sec^6(x) dx = (1/5) * sec^4(x) * tan(x) + (4/5) * ∫ sec^4(x) dx

Step 3: Apply the reduction formula again to the remaining integral: ∫ sec^4(x) dx = (1/3) * sec^2(x) * tan(x) + (2/3) * ∫ sec^2(x) dx

Step 4: Simplify the expression: ∫ sec^4(x) dx = (1/3) * sec^2(x) * tan(x) + (2/3) * tan(x) + C

Step 5: Substitute the simplified expression back into the original integral: ∫ sec^6(x) dx = (1/5) * sec^4(x) * tan(x) + (4/5) * ((1/3) * sec^2(x) * tan(x) + (2/3) * tan(x) + C)

Step 6: Simplify the expression further if needed.

II. ∫ sin^4(x) cos^5(x) dx

To solve this integral using the reduction formula, we need to express sin^4(x) in terms of cos(x) using the identity sin^2(x) = 1 - cos^2(x).

Step 1: Rewrite sin^4(x) as (sin^2(x))^2: sin^4(x) = (1 - cos^2(x))^2

Step 2: Expand the expression: sin^4(x) = 1 - 2cos^2(x) + cos^4(x)

Step 3: Substitute the expression for sin^4(x) into the integral: ∫ sin^4(x) cos^5(x) dx = ∫ (1 - 2cos^2(x) + cos^4(x)) cos^5(x) dx

Step 4: Simplify the expression: ∫ sin^4(x) cos^5(x) dx = ∫ cos^5(x) dx - 2∫ cos^7(x) dx + ∫ cos^9(x) dx

Step 5: Apply the reduction formula to each integral: ∫ cos^5(x) dx = (1/5) * cos^4(x) * sin(x) + (4/5) * ∫ cos^3(x) dx ∫ cos^7(x) dx = (1/7) * cos^6(x) * sin(x) + (6/7) * ∫ cos^5(x) dx ∫ cos^9(x) dx = (1/9) * cos^8(x) * sin(x) + (8/9) * ∫ cos^7(x) dx

Step 6: Substitute the simplified expressions back into the original integral: ∫ sin^4(x) cos^5(x) dx = (1/5) * cos^4(x) * sin(x) + (4/5) * ((1/7) * cos^6(x) * sin(x) + (6/7) * ((1/5) * cos^4(x) * sin(x) + (4/5) * ∫ cos^3(x) dx)) + (1/9) * cos^8(x) * sin(x) + (8/9) * ((1/7) * cos^6(x) * sin(x) + (6/7) * ((1/5) * cos^4(x) * sin(x) + (4/5) * ∫ cos^3(x) dx)) + C

Step 7: Simplify the expression further if needed.

Please note that these are the general steps to solve the given integrals using the reduction formula. The final expressions may vary depending on the specific values of x and the limits of integration.