7.1 Integration By Parts/29: Difference between revisions
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<math> f'(x)= \int_{}^{}\cos(x)\ln(sin(x))\cdot dx </math> <br><br> | <math> f'(x)= \int_{}^{}\cos(x)\ln(sin(x))\cdot dx </math> <br><br> | ||
<math>\int_{}^{}\cos(x)\ln(\sin(x))\cdot dx=\int_{}^{}\ln(u)\cdot du</math><br> | |||
<math>u=\sin(x)</math> <br> <math> du=\cos(x)dx | <math>\int_{}^{}\cos(x)\ln(\sin(x))\cdot dx = \int_{}^{}\ln(u)\cdot du ~ ~ ~ = ~ ~ ~ ~ uln(u)-\int_{}^{} du =u \cdot ln(u)-u+c</math><br> | ||
<math>u=\sin(x) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ z=\ln(u) ~ ~ ~ ~ ~ dw=du</math> <br> | |||
<math> du=\cos(x)\cdot dx ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ dz=\frac{1}{x}\cdot du ~ ~ w=u</math> <br><br> | |||
<math>\text{Therefore, } f(x)=sin(x)ln(sin(x))-sin(x)+c</math> | <math>\text{Therefore, } f(x)=sin(x)ln(sin(x))-sin(x)+c</math> | ||
Revision as of 22:18, 26 November 2022
