7.1 Integration By Parts/32: Difference between revisions
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<math>\int_{0}^{t} e^s \sin(t-s) \cdot ds ~ ~ ~ = ~ ~ ~ e^s \cdot \cos(t-s) - \int_{0}^{t} e^s \cos(t-s) \cdot ds ~ ~ ~ = ~ ~ ~ e^s \cdot \cos(t-s) - e^s \cdot \sin(t-s) -\int_{0}^{t} e^s \sin(t-s) \cdot ds</math><br> | <math>\int_{0}^{t} e^s \sin(t-s) \cdot ds ~ ~ ~ = ~ ~ ~ e^s \cdot \cos(t-s) - \int_{0}^{t} e^s \cos(t-s) \cdot ds ~ ~ ~ = ~ ~ ~ e^s \cdot \cos(t-s) - e^s \cdot \sin(t-s) -\int_{0}^{t} e^s \sin(t-s) \cdot ds</math><br> | ||
<math> u= e^s ~ ~ ~ ~ ~ dv=\sin(t-s)dx ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ u=e^s ~ ~ ~ ~ ~ dv=\cos(t-s)</math><br> | <math> u= e^s ~ ~ ~ ~ ~ dv=\sin(t-s)dx ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ u=e^s ~ ~ ~ ~ ~ dv=\cos(t-s)</math><br> | ||
<math>du= e^s ds~ ~ v=\cos(t-s) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ du=e^s ~ ~ ~ ~ ~ v=-\sin(t-s)</math> <br><br> | <math>du= e^s ds~ ~ v=\cos(t-s) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ du=e^s ~ ~ ~ ~ ~ v=-\sin(t-s)</math> <br><br> | ||
Latest revision as of 17:06, 29 November 2022
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f'(x)= \int_{0}^{t} e^s sin(t-s) \cdot ds }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_{0}^{t} e^s \sin(t-s) \cdot ds ~ ~ ~ = ~ ~ ~ e^s \cdot \cos(t-s) - \int_{0}^{t} e^s \cos(t-s) \cdot ds ~ ~ ~ = ~ ~ ~ e^s \cdot \cos(t-s) - e^s \cdot \sin(t-s) -\int_{0}^{t} e^s \sin(t-s) \cdot ds}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u= e^s ~ ~ ~ ~ ~ dv=\sin(t-s)dx ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ u=e^s ~ ~ ~ ~ ~ dv=\cos(t-s)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle du= e^s ds~ ~ v=\cos(t-s) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ du=e^s ~ ~ ~ ~ ~ v=-\sin(t-s)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_{0}^{t} e^s \sin(t-s) \cdot ds ~ ~ ~ = ~ ~ ~ e^s \cdot \cos(t-s) - e^s \cdot \sin(t-s) -\int_{0}^{t} e^s \sin(t-s) \cdot ds}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2\int_{0}^{t} e^s \sin(t-s) \cdot ds ~ ~ ~ = ~ ~ ~ e^s \cdot \cos(t-s) - e^s \cdot \sin(t-s)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \int_{0}^{t} e^s \sin(t-s) \cdot ds ~ ~ ~ = ~ ~ ~ \frac{e^s}{2}(\cos(t-s) - \sin(t-s)) \Bigg|_0^t}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =\frac{e^t}{2}(\cos(0) - \sin(0)) -\frac{e^0}{2}(\cos(t) - \sin(t))}
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