Question

calcule a área da região limitada pelas curvas y=x, y=x/4 e y=1/x

Answer 1

$\int {x^n}\ dx=\frac{x^{n+1}}{n+1}+C\\\\\int {\frac{1}{x}}\ dx=\ln|x|+C\\\\\int{c.f(x) \dx=c.\int{f(x)\ dx$

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$\int\limits^1_0 {x-\frac{x}{4}} \, dx +\int\limits^2_1 {\frac{1}{x}-\frac{x}{4}} \, dx \\\\\int\limits^1_0 x} \, dx -\frac{1}{4}\int\limits^1_0 {x} \, dx+\int\limits^2_1 {\frac{1}{x}} \, dx -\frac{1}{4}\int\limits^2_1 {x} \, dx$

$[(\frac{1^2}{2})-(\frac{0^2}{2})]-\frac{1}{4}[(\frac{1^2}{2})-(\frac{0^2}{2})]+[(\ln2)-(\ln1)]-\frac{1}{4}[(\frac{2^2}{2})-(\frac{1^2}{2})]\\\\\frac{1}{2}-\frac{1}{8}+\ln2-\frac{3}{8}\\\\\ln2$