- 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*). - Select a value for
*a*-- this can be done either of two ways:- Click and drag the red point in the
*left*graph. The point moves along the*x*-axis, and its horizontal position gives the value for*a*. - Enter a value directly into the text input field marked
"
*a*=".

*a*=1) - Click and drag the red point in the
- Enter a value for the
*base*length of the rectangle shown in the*left*graph in the text input field marked "base=". (Example:*b*=1) - To erase the graph and all input fields, setting
*a*and the*base*width to default values, click the "Clear" button

For assistance computing the derivative *g*'(*x*), try
the Derivative Calculator.

- Basic Example:

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

∫*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*=0,*base*=1

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

*f*(*x*)=1,*u*= sin*x*,*u*'= cos*x*,*a*=0,*base*=0.5

∫*f*(*x*)*dx*= ∫(1/√(1-u^2))*du*

The graphs show a function

The graph on the left also shows a green
or orange
rectangle with a red point at one corner.
The height of the rectangle is the value of *f* (*a*), where
*a* is the horizontal position of the red point.
This value of *f* (*a*) is given in a label under the graph.
The base width of the rectangle is given in the text input field
marked "base=".
If the area of the rectangle is positive, it is shaded in
green; if negative, orange.
The area of the rectangle is also shown in a label under the graph.

A corresponding green
or orange
rectangle and red point are drawn in the graph on
the right. The horizontal position of the red
point is given by *g*(*a*). The height of the rectangle
is given by *f* (*a*)/*g*'(*a*) (the point on
the left graph over the red point).
The base width of the rectangle is *b*^{.}*g*'(*a*),
where *b* is the base width from the left graph.
If the area of the rectangle is positive, it is shaded in
green; if negative, orange.
Labels under the right graph show *g*(*a*), the base width
and height of the rectangle, and the area of the rectangle.

The areas of the rectangles in the two graphs are equal. With the two
graphs giving different heights, and the base width in the left graph
fixed by the text input field, the critical dimension is the base
width in the right graph -- this is the effect of the substitution
*u*=*g*(*x*).