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 (iii)
Usually, if any function is a power of x or a polynomial in x, then we take it as the
first function