For instance, the method does not work for 
xsin
x dx
∫ The reason is that there does not exist any function whose derivative is
x  sin x (ii)
Observe that while finding the integral of the second function, we did not add
any constant of integration If we write the integral of the second function cos x
INTEGRALS         325
as sin x + k, where k is any constant, then
xcos
x dx
∫
 =
(sin
)
(sin
)
x
x
k
x
k dx
+
−
+
∫
=
(sin
)
(sin
x
x
k
x dx
k dx
+
−
−
∫
∫
=
(sin
)
cos
C
x
x
k
x – kx
+
−
+
 = 
sin
cos
C
x
x
x
+
+
This shows that adding a constant to the integral of the second function is
superfluous so far as the final result is concerned while applying the method of
integration by parts