|
| Integral
calculus |
|
|
|
The
indefinite integral |
Integration by parts rule |
|
The indefinite integrals
containing quadratic polynomial (trinomial)
|
|
The indefinite integrals
containing quadratic polynomial, examples
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Integration
by parts rule
|
|
The rule for differentiating the
product of two differentiable functions leads to the integration
by parts formula.
|
|
Let f
(x)
and g
(x)
are differentiable functions, then the product rule gives
|
|
[ f
(x) g (x)]'
= f (x)
g (x)' + g (x)
f ' (x),
|
| by
integrating both sides |
|
|
Since the integral of the derivative
of a function is the function itself, then
|
|
|
and by rearranging obtained is
|
|
|
the integration by parts
formula.
|
|
By substituting
u = f
(x) and
v
= g (x)
then, du =
f ' (x)
dx and dv
= g' (x)
dx, so that
|
|
|
To apply the above formula, the
integrand of a given integral should represent the product of one
function and
|
|
the differential of the other.
|
|
The selection of the function
u and the differential dv
should simplify the
evaluation of the remaining integral.
|
|
In some cases it will be necessary to
apply the integration by parts repeatedly to obtain a simpler
integral.
|
|
|
|
The indefinite integrals
containing quadratic polynomial (trinomial)
|
| Recall
that the quadratic polynomial y
= ax2 + bx + c
represents the expansion of the translatable form of |
|
its source function y
= ax2 in the direction of the coordinate axes, thus
|
|
|
|
|
|
| a
)
Solution:
if b2
-
4ac < 0
substitutions,
|
|
|
 |
|
|
|
|
|
| b
)
Solution:
if b2
-
4ac > 0
substitutions,
|
|
|
 |
|
|
|
|
|
| c
) Solution:
if b2
-
4ac = 0 then
k
= 0, substitution
|
 |
gives |
|
|
|
|
|
|
The indefinite integrals
containing quadratic polynomial examples
|
| Example:
53.
a
)
Evaluate |
 |
|
|
|
|
|
| Example:
53.
b
)
Evaluate |
| |
|
 |
|
|
|
|
|
| Example:
53.
c
)
Evaluate |
|
|
 |
|
|
|
|
|
|
|
|
Since the numerator of the integrand
is the derivative of the denominator therefore, the substitution
|
|
t = ax2
+ bx + c and
dt =
(2ax + b) dx
gives
|
|
|
|
|
|
| by substituting
in the numerator |
 |
|
|
|
|
|
| Example:
54.
a
)
Evaluate |
|
|
 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| Functions
contents G |
|
|
 |
|
| Copyright
© 2004 - 2020, Nabla Ltd. All rights reserved. |
