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 However, in cases where other function is inverse trigonometric
function or logarithmic function, then we take them as first function Example 18 Find log x dx
∫
Solution To start with, we are unable to guess a function whose derivative is log x