To integrate the forms

Xdx sin.~' X, (I)

Xc?a! COS."' X, (2)

in which X is an algebraic function of a;
1°. Let ■M=sin.~' x/

dx

VI— a;*
dv=X<h:; .:v=zfXdx. ,
Then fudv=uv—/vdu

becomes fX.dx sin.~'a;=sin.~'a!/'Xc?a;— I , .(jb.
•'J Vl—x^
Formula M.
Ex. 111. Integrate . sin.~' x.
^ -Vl—x^
a?dx
First integrate . by Formula A and Art. 187,

VI— a;2

and we have

/* x^dx r

/dx
I , we have
VI— a;^

"9+TJ-

Substituting these in the above formula, we have
a?dx



k



■\/\—x^
— C^^+DVi— a:^ sin.-' x+^x^+^x^+G.
Integeation between Limits,

208. In the process of differentiation, we saw that all
the constants which were added to or subtracted from
the variable quantities disappeared ; arid, on the contrary,
when we performed an integration, we added a constant
quantity to correct it, which we represented by the let-
ter C. In order to find the value of this constant, we



EECTIFICATION OP PLANE CUEVES. 275

must first find what particular value of the variable makes
the integral equal to zero; we then have two equations
each containing*C, which quantity may be eliminated
by taking the difierence between the two equations.
Thus, if we have /2a;(?a;=a!^+C for the general value
of the integral, and the problem indicates that, for the
particular value, x=a, the integral becomes zero; then
0=a^+G. If we subtract this from the general value
of the integral, we will have

and the constant C has disappeared. This is called the
corrected integral, and is expressed thus :



/



2xdx=.x'^—a'^.
The value of this integral commences when x=.a.
Now, if we give another value to x, say a:=5,then we
have fully determined the value of the integral, and it is
written

'ixdx=1P~a'^.
r.
