Mathematics Times – July 2019

^

/ 4

2

4

1 1

2 (2 cos 1)

I dx

x







  

/4 2

2

4

1 sec

3sec 2

x

dx

x







 

put tanx t implies that sec^2 xdx dt

rewriting the above integral interms of ‘t’ we

get

WHAT IS THERE TO KNOW?

There are many “cute” little numerical tricks for performing calculations quickly.

For example, here‘s a little trick to multiply two positive integers with same tens digits and

with the units digits summing to ten.

TRICK
Multiply the tens digit by the next largest number. Call your results “A”,
Then multiply the units digits together, and call your result “B”,
 Write A immediately followed by B ( where B is considered a two-digit number), and read it as a
single number,
 You have your product!

For example, to multiply 32 and 38 in your head. you get

1 1

2

(^00)
2 2 1
tan
(^3131) 1 / 3
3
dt t
I
 t 
    
  
    
 
 



1 1

(^2) (tan ( 3) tan (0)) 2 2
3 3 3 3 3

 
      
  
Now
27 2 27 4 4
27
I   
A (^12) and B = 16, So 32 38 1216 .
Proof: Let x be the tens digit of both numbers, and y be units digit of the first number. Then 10 - y is
the units digit of the second number. Then two numbers are therefore 10x + y and 10x + 10
TRICK




Proof:
y.
Multiplying then together, we get
(10x y    ) (10x 10 y) 100x^2  100 x y y(10 )
100 (x x 1) y y(10 )
 100 A B
A x x B y y  ( 1); (10 )