### Substitution 1

Graphs a function

*f* (

*x*), the area under the graph of

*f* (

*x*) for a given interval,
and the modifications made to

*f* (

*x*) and the area by a
substitution

*u*=

*g*(

*x*)

How to use
||
Examples
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Other Notes

**How to use**

- Enter the function
*f* (*x*) in the text
input field marked "*f* (*x*)="
(Example:
*f* (*x*)=2*x*/(*x*^{2}+1)).
- Click the "Graph" button (this button refreshes
both graphs) -- at this point, the
*left* graph will show the
graph of *f* (*x*), and the *right* graph will remain
empty.
- Enter the substitution function
*g*(*x*) in the text
input field marked "*u*=*g*(*x*)=",
and its derivative *g*'(*x*) in
the text input field marked "*u*'=*g*'(*x*)=".
Then click the "Graph" button to refresh.
Now the *right* graph will show the function transformed by the
substitution.
(Example:
*u*=*x*^{2}+1,
*u*'=2*x*).
- Enter the endpoints of the interval [
*a*,*b*]
for the definite integral in the text input fields marked
"*a*=" and "*b*=", and click
the "Graph" button to refresh.
(Example:
[*a*,*b*]=[1,2]).
- To erase the graph and all input fields, setting
*a* and *b* to default values, click the
"Clear" button

The text input fields for *f* (*x*), *g*(*x*),
and *g*'(*x*) can accept
a wide variety of expressions to represent functions, and the
buttons under the graph
allow various manipulations of the graph coordinates.
The text input fields for *a* and *b* can accept real numbers
in decimal notation.
For assistance computing the derivative *g*'(*x*), try
the Derivative Calculator.

**Examples**

- Basic Example:

*f* (*x*)=2*x*/(*x*^{2}+1),
*u*=*x*^{2}+1, *u*'=2*x*,
[*a*,*b*]=[1,2]

∫*f*(*x*) *dx*
= ∫(1/*u*)*du*
- Trigonometric Substitution -- Forward:

*f* (*x*)=1/√(1-*x*^{2}),
*u*= sin^{-1} *x*,
*u*'=1/√(1-*x*^{2}),
[*a*,*b*]=[0,0.8]

∫*f*(*x*) *dx*
= ∫1 *du*
- Trigonometric Substitution -- Backward:

*f* (*x*)=1,
*u*= sin *x*,
*u*'= cos *x*,
[*a*,*b*]=[0,0.9]

∫*f*(*x*) *dx*
= ∫(1/√(1-u^{2}))*du*

**Other Notes**

The area under the graph of
*f* (*x*) is drawn
with the area under the curve on the interval
[*a*,*b*] shaded so that positive areas are
blue or green
and negative areas are red
or orange.
The graph on the left shows the areas after making the substitution
*u*=*g*(*x*), so plotting
(*g*(*x*), *f* (*x*)/*g*'(*x*))
for *x* in the interval [*a*,*b*].
Correspondingly colored regions in the two graphs
have equal area.
Note that green and orange
regions in the right graph have opposite orientation to their
counterparts in the left graph.
The endpoints of the interval in the right graph are given as
*c*=*g*(*a*) and *d*=*g*(*b*)
-- the values of *c* and *d*
are shown in labels under the graph.
Be aware that regions in the left
graph may overdraw (for example, a blue region
may overdraw an orange region) --
regions are overdrawn as though they are semitransparent.

Another view of Substitution