User:Jw116104

#1
Find $$I = \int_1^2 x\ln{x}\, \mathrm dx\!$$.

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

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

So,

$$I=uv\right|_1^2-\int_1^2 v\,\mathrm du.\!$$

#2
Find $$I = \int (x+\ln{x}+e^x)\,\mathrm dx\!$$.

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

$$I = (\frac{x^2}{2}+c_1) + \int\ln{x}\,\mathrm dx\! + (e^x +c_3)$$

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

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

So,

$$I = (\frac{x^2}{2}+c_1) + uv-\int v\,\mathrm du\! + (e^x +c_3)$$

$$I = (\frac{x^2}{2}+c_1) + (\ln{x})(x)-\int (x)\,(\frac{1}{x}\mathrm dx\!) + (e^x +c_3)$$

$$I = (\frac{x^2}{2}+c_1) + (\ln{x})(x)-\int \mathrm dx\! + (e^x +c_3)$$

$$I = (\frac{x^2}{2}+c_1) + (x\ln{x}-x+c_2) + (e^x +c_3)$$

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