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}
If m is even and n is even
Here m is odd or even and n is odd, then
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 sin^{2}x
sin^{2}x = 1 - cos^{2}x
sin^{2}x cos^{4}x = (1 - cos^{2}x) (cos^{4}x)
= cos^{4}x - cos^{6}x
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 :
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM