Reverse the product rule
Integrating the product rule and rearranging gives integration by parts: integrate one factor and differentiate the other. Keep the boundary term or constant appropriate to the integral.
Calculus / English
Integration by parts (visualised)
Mathacy · YouTube · 8:53
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Derive integration by parts from the product rule and connect it to a geometric area identity and worked examples.
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Chapters
0:30Reverse the product rule3:00Two areas and a boundary term6:00Reduce the polynomial degreeLearning script
Generated from the video's visuals and explanation; not verbatim speech.
Integrating the product rule and rearranging gives integration by parts: integrate one factor and differentiate the other. Keep the boundary term or constant appropriate to the integral.
The visualization combines an area integrated with respect to x and another with respect to y into a difference of endpoint rectangles. Parametric substitution leads to the definite formula.
For x²eˣ, differentiating the polynomial reduces its degree, so applying integration by parts twice finishes the calculation. For ln x, treat the other factor as 1 on .
Knowledge cards
Two areas and a boundary term
The visualization combines an area integrated with respect to x and another with respect to y into a difference of endpoint rectangles. Parametric substitution leads to the definite formula.
Reduce the polynomial degree
For x²eˣ, differentiating the polynomial reduces its degree, so applying integration by parts twice finishes the calculation. For ln x, treat the other factor as 1 on .
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- Integration by parts ExplanationAt 0:30
Why this connection?
Integrating the product rule and rearranging gives integration by parts: integrate one factor and differentiate the other. Keep the boundary term or constant appropriate to the integral. The visualization combines an area integrated with respect to x and another with respect to y into a difference of endpoint rectangles. Parametric substitution leads to the definite formula. For x²eˣ, differentiating the polynomial reduces its degree, so applying integration by parts twice finishes the calculation. For ln x, treat the other factor as 1 on .
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Connect the next step
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A different approach
YouTube8:34Lecture for Integration by parts, compared with the current visual explanation.
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Whiteboard for Integration by parts, compared with the current visual explanation.
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YouTube17:59Whiteboard for Integration by parts, compared with the current visual explanation.
Integration by parts