User:Jw116104

Math 163b

#1

Find $I=\int _{1}^{2}x\ln {x}\,\mathrm {d} x\!$.

Let $u=\ln {x}$ and $dv=x\,\mathrm {d} x\!$.

Then, $du={\frac {1}{x}}\,\mathrm {d} x\!$ and $v={\frac {x^{2}}{2}}$.

So,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/wiki.riteme.site/v1/":): {\displaystyle I=uv\right|_1^2-\int_1^2 v\,\mathrm du.\!}

#2

Find $I=\int (x+\ln {x}+e^{x})\,\mathrm {d} x\!$.

$I=\int x\,\mathrm {d} x\!+\int \ln {x}\,\mathrm {d} x\!+\int e^{x}\,\mathrm {d} x\!$

$I=({\frac {x^{2}}{2}}+c_{1})+\int \ln {x}\,\mathrm {d} x\!+(e^{x}+c_{3})$

Let $u=\ln {x}$ and $dv=\mathrm {d} x\!$.

Then, $du={\frac {1}{x}}\mathrm {d} x\!$ and $v=x$.

So,

$I=({\frac {x^{2}}{2}}+c_{1})+uv-\int v\,\mathrm {d} u\!+(e^{x}+c_{3})$

$I=({\frac {x^{2}}{2}}+c_{1})+(\ln {x})(x)-\int (x)\,({\frac {1}{x}}\mathrm {d} x\!)+(e^{x}+c_{3})$

$I=({\frac {x^{2}}{2}}+c_{1})+(\ln {x})(x)-\int \mathrm {d} x\!+(e^{x}+c_{3})$

$I=({\frac {x^{2}}{2}}+c_{1})+(x\ln {x}-x+c_{2})+(e^{x}+c_{3})$

$I={\frac {x^{2}}{2}}+x\ln {x}-x+e^{x}+C$