REDUCTION FORMULA FOR SIN TO THE POWER OF M X AND COS TO THE POWER N X

The method of obtaining reduction formula has the following steps :

Step 1 :

Identify an index (positive integer) n in the integral

Step 2 :

Put the integral as I_n

Step 3 :

Applying integration by parts, obtain the equation for I_n in terms of I_{n -1} or I _{n - 2}

The resulting equation is called reduction formula for I_n

Evaluate the following :

Problem 1 :

Solution :

Since these two trigonometric ratios are multiplied to make it as once, we can use the formula for sin²x

sin²x = 1 - cos²x

sin²x  cos⁴x = (1 - cos²x) (cos⁴x)

= cos⁴x - cos⁶x

Problem 2 :

Solution :

Here m = 3 and n = 5

m may be odd or even, since n is odd, we use the formula

Problem 3 :

Solution :

Problem 4 :

Solution :