Article 26
Elementary Integrals.
The following useful indefinite integrals follow from Arts. 14, 15, 23:
| Hyperbolic. | Circular.
|
1. | | |
2. | | |
3. | | |
4. | | |
5. | | |
| | |
6. | | |
7. | 4 | |
8. | | |
9. | | |
10. | | |
11. | | |
12. | | |
From these fundamental integrals the following may be derived:
|
|
Thus,
(By interpreting these two integrals as areas, show graphically that the first is
positive, and the second negative.)
the real form to be taken. (Put ,
and apply 9, 10.)
the real form to be taken.
|
By means of a reduction-formula this integral is easily made to depend on 8. It
may also be obtained by transforming the expression into hyperbolic functions by the
assumption ,
when the integral takes the form
which gives 17 on replacing
by ,
and
by .
The geometrical interpretation of the result is evident, as it
expresses that the area of a rectangular-hyperbolic segment
is the difference
between a triangle
and a sector .
-
Prob. 71.
- What is the geometrical interpretation of 18, 19?
-
Prob. 72.
- Show that
reduces to 17, 18, 19, respectively: when
is positive, with ;
when
is negative; and when
is positive, with .
-
Prob. 73.
- Prove
-
Prob. 74.
- Integrate ,
,
.
-
Prob. 75.
- In the parabola ,
if
be the length of arc measured from the vertex, and
the angle
which the tangent line makes with the vertical tangent, prove that the intrinsic equation of the
curve is ,
.
-
Prob. 76.
- The polar equation of a parabola being
, referred to its focus
as pole, express
in terms of .
-
Prob. 77.
- Find the intrinsic equation of the curve
, and of
the curve .
-
Prob. 78.
- Investigate a formula of reduction for
;
also integrate by parts
,
,
;
and show that the ordinary methods of reduction for
can be
applied to .