Question

Calcule as integrais:

a)∫( x^5-2X²+3X/x²)dx

B)∫(x^5-6x³+3senx)dx

Answer 1

A integral da soma/diferença é a soma/diferença das integrais:

$\boxed{\boxed{\int(a(x)\pm b(x)\pm...)dx=\int a(x)dx\pm\int b(x)dx\pm...}}$

Integral de potências:

$\boxed{\boxed{\int t^{n}dt=\dfrac{t^{n+1}}{n+1}+C~~~\forall~n\neq-1,~~~pois~\dfrac{d}{dt}\left(\dfrac{t^{n+1}}{n+1}+C\right)=t^{n}}}\\\\\\\boxed{\boxed{\int\dfrac{1}{t}dt=ln|t|+C,~~~pois~\dfrac{d}{dt}(ln|t|+C)=\dfrac{1}{t}}}$
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A)

$\displaystyle\int\dfrac{x^{5}-2x^{2}+3x}{x^{2}}dx=\int\left(\dfrac{x^{5}}{x^{2}}-\dfrac{2x^{2}}{x^{2}}+\dfrac{3x}{x^{2}}\right)dx\\\\\\\int\dfrac{x^{5}-2x^{2}+3x}{x^{2}}dx=\int\left(x^{3}-2x^{0}+\dfrac{3}{x}\right)dx\\\\\\\int\dfrac{x^{5}-2x^{2}+3x}{x^{2}}dx=\int x^{3}dx-\int2x^{0}dx+\int\dfrac{3}{x}dx\\\\\\\int\dfrac{x^{5}-2x^{2}+3x}{x^{2}}dx=\dfrac{x^{4}}{4}-2\int x^{0}dx+3\int\dfrac{1}{x}dx\\\\\\\int\dfrac{x^{5}-2x^{2}+3x}{x^{2}}dx=\dfrac{x^{4}}{4}-2\dfrac{x^{0+1}}{0+1}+3ln|x|+constante$

$\boxed{\boxed{\int\dfrac{x^{5}-2x^{2}+3x}{x^{2}}dx=\dfrac{x^{4}}{4}-2x+3ln|x|+constante}}$

B)

$\displaystyle\int(x^{5}-6x^{3}+3sen~x)dx=\int x^{5}dx-\int6x^{3}dx+\int3sen(x)dx\\\\\\\int(x^{5}-6x^{3}+3sen~x)dx=\dfrac{x^{5+1}}{5+1}-6\int x^{3}dx+3\int sen(x)dx\\\\\\\int(x^{5}-6x^{3}+3sen~x)dx=\dfrac{x^{6}}{6}-6\dfrac{x^{3+1}}{3+1}+3(-cos(x))+consante\\\\\\\int(x^{5}-6x^{3}+3sen~x)dx=\dfrac{x^{6}}{6}-\dfrac{6x^{4}}{4}-3cos(x)+constante\\\\\\\boxed{\boxed{\int(x^{5}-6x^{3}+3sen~x)dx=\dfrac{x^{6}}{6}+\dfrac{3}{2}x^{4}-3cos(x)+constante}}$