Example.1
If for non-zero x, af(x) + b f
=
where
a > b, then find
.
Solution
af(x) + b f
=
….
(1)
Integrating, we have
a
…. (2)
Replacing x by
in (1), we get, af
+ bf(x) = x –5
Integrating, we
have, a
…. (3)
Eliminating
from (2) and (3) by multiplying (2) by
a and (3) by b and subtracting, we get (a2 – b2) ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image010.gif)
&⇒
.
Example.2
Prove that
=
2p.
Solution
Let I = real ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image013.gif)
= real part of
= real part ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image015.gif)
= real part ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image016.gif)
= real part ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image017.gif)
= real part ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image018.gif)
=![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image019.gif)
= ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image020.gif)
= (2p + 0 + 0 + .....) – (0 + 0 + 0
+.....) = 2p.
Example.3
Evaluate ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image021.gif)
Solution
Let tan-1x
= q. As
, so –![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image023.gif)
Now ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image024.gif)
= ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image025.gif)
= cos-1(
sin2q) + tan-1(
tan2q) = cos-1
+ tan-1( tan2q)
=
– 2q
+ 2q [ as
and
]
= p/2
Hence, I = ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image030.gif)
=
=
=
.
Example.4
If Un
=
where n is a positive integer or zero,
then show that Un+2 + Un = 2 Un+1. Hence
deduce that ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image035.gif)
Solution
Un+2 – Un+1 =
![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image036.gif)
=
= ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image038.gif)
Un+2 –
Un+1 = ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image039.gif)
Un+1 – Un =
From (1) and (2), we get
(Un+2 – Un+1) – (Un+1
– Un) = ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image042.gif)
Un+2 + Un – 2Un+1
= ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image043.gif)
=
=
= 0
Un+2 + Un = 2 Un+1
Un, Un+1, Un+2
are in A.P.
Now, U0 =
=
=0,
U1 =
U1 – U0 =
(Common difference)
Un = U0 + n. ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image049.gif)
Un = n![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image049.gif)
Now In =
(
writing 2q = x )
=
= ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image052.gif)
Alternative:
We have Un+2
+ Un = ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image053.gif)
= 2
= ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image055.gif)
= 2Un+1 +
0. .
Example.5
Find a
function g: R → R, continuous in [0, ¥)
and positive in (0, ¥) satisfying g (1) =1 and
.
Solution
Let F (x) =
![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image057.gif)
![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image058.gif)
![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image059.gif)
![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image060.gif)
![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image061.gif)
Differentiating both
sides with respect to x, we get
![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image062.gif)
![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image063.gif)
![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image064.gif)
&⇒
t2 – 4 t + 2 = 0 where ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image065.gif)
&⇒
or ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image067.gif)
&⇒
ln F (x) = (2
) lnx + constant
&⇒
F (x) = C ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image069.gif)
&⇒
![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image070.gif)
where
are
constants. But g is continuous on [0, ¥).
Then c²
is ruled out.
Hence g (x) = ![](http://www.quizsolver.com/radix/dth/notif/Definite%20SUB%204_files/image073.gif)
Also g (1) =
= 1
.