Integration by Parts

Graphs a function f (x)=g(x)h'(x) and the area under the graph of f (x) for a given interval, and shows the modifications made to f (x) and the area when considering u=g(x) and v=h(x) as independent variables, as when carrying out the integral using the technique of Integration by Parts
How to use   ||   Examples   ||   Other Notes

How to Use
• Enter the function g(x) in the text input field marked "u=g(x)="
(Example: g(x)=x -- no graph yet)
• Enter the function h'(x) in the text input field marked "v'=h'(x)="
(Example: h'(x)=ex -- graph on left)
• Enter the antiderivative h(x) of h'(x) in the text input field marked "v=h(x)="
(Example: h(x)=ex -- both graphs)
• Enter the endpoints of the interval [a,b] for the definite integral in the text input fields marked "a=" and "b=" (Example: [a,b]=[0.5,1.5] -- areas shaded)
• Click the "Graph" button (this button also refreshes the graph)
• To erase the graph and all input fields, click the "Clear" button
The text input fields marked "u=g(x)=", "v'=h'(x)=", and "v=h(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 marked "a=" and "b=" can accept real numbers.

For assistance checking the derivative h'(x), try the Derivative Calculator.

Examples
 Polynomial and Exponential: f (x)=x ex (g(x)=x, h'(x)=ex) [a,b]=[0.5,1.5] Polynomial and Trigonometric: f (x)=x cos(x) (g(x)=x, h'(x)=cos(x)) [a,b]=[0,Pi/2] Exponential and Trigonometric: f (x)=ex cos(x) (g(x)=ex, h'(x)=cos(x)) [a,b]=[0,Pi/2] Logarithmic: f (x)=ln(x) (g(x)=ln(x), h'(x)=1) [a,b]=[2,6]

Other Notes
The graph of f (x)=g(x).h'(x) is shown in the left graph, with the area under the curve on the interval [a,b] shaded so that positive areas are blue and negative areas are red.

In the right graph, the curve graphed is given by (u,v)=(g(x),h(x)) for x in [a,b], with areas between the curve and the vertical v axis colored blue and red as in the right graph (but oriented differently), and areas between the curve and the horizontal u axis colored green (for positive areas) and orange (for negative areas).

Values of c=g(a) and d=g(b) (interval endpoints on the u-axis), and e=h(a) and f=h(b) (interval endpoints on the v-axis) are also shown -- the total net area of the colored regions in the right graph will be df-ce.

Moreover, the corresponding blue and red regions will have the same area in the two graphs.

Recall the formula for Integration by Parts: ∫u dv = uv - ∫v du

In the "Polynomial and Exponential" example above, the overall shaded region in the right graph appears as an inverted "L" shape, from a large rectangular region (with area df) minus a smaller rectangular region (with area ce). This corresponds to the uv term of the Integration by Parts formula, evaluated with the endpoints a and b. The area shaded green in the example corresponds to the integral ∫v du (v on vertical axis with respect to u on horizontal axis). Subtracting this region from the "L" shaped region leaves only the blue region in the right graph. This blue region corresponds to the integral ∫u dv (u with respect to v), and has the same area as the region shaded blue in the left graph.

Be aware that regions in the right graph may overdraw (for example, a blue region may overdraw an orange region) -- in this case, the overdrawn regions combine colors, but the resulting combinations are somewhat unpredictable, and may even include solid black or white regions. The endpoints of the intervals in the examples above have been carefully chosen to avoid this possibility.

Update: With the upgrade to the Java2 platform for this applet, the behavior in the right graph when regions overdraw is different -- the regions overdraw as if they are simply semitransparent layers. This makes the color combinations a bit more predictable.