REDUCTION FORMULA FOR DEFINITE INTEGRALS

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 :

Problem 2 :

Solution :

Problem 3 :

Solution :

Here the limit is not 0 to 𝜋/2, so let us use the substitution method to convert the limits as 0 to 𝜋/2.

Let 2x = t

2dx = dt

dx = dt/2

When x = 0, t = 0

When x = 𝜋/4, t = 2(𝜋/4) ==> 𝜋/2

Problem 4 :

Solution :

Here the limit is not 0 to 𝜋/2, so let us use the substitution method to convert the limits as 0 to 𝜋/2.

Let 3x = t

3 dx = dt

dx = dt/3

When x = 0, t = 0

When x = 𝜋/6, t = 3(𝜋/6) ==> 𝜋/2

Problem 5 :

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 6 :

Solution :

Here the limit is not 0 to 𝜋/2, so let us use the substitution method to convert the limits as 0 to 𝜋/2.

Let x/4 = t

dx = 4dt

When x = 0, t = 0

When x = 2𝜋, t = (2𝜋/4) ==> 𝜋/2

Problem 7 :

Solution :

Here m = 3 and n = 5

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

Problem 8 :

Solution :