Question

Calcule as integrais. i) ∫ (lnx) 3/ x dx ii) ∫ e x 4 ⋅ x 3 dx

Answer 1

Integral Indefinida;

$f \: x^{4}x^{3} dx \\ \\ f \: x^{7} dx \\ \\ \frac{x^{7 + 1} }{7 + 1} = \frac{x^{8} }{8}$

Obs; Propriedade do tipo;

f (t^n) = t^(n + 1)/(n + 1)

Answer 2

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$\displaystyle\sf\int\dfrac{(\ell n x)^3}{x}~dx\\\rm fac_{\!\!,}a~t=\ell nx\implies dt=\dfrac{1}{x}dx\\\displaystyle\sf\int\dfrac{(\ell nx)^3}{x}dx=\int t^3~dt=\dfrac{1}{4}t^4+k\\\huge\boxed{\boxed{\boxed{\boxed{\displaystyle\sf\int\dfrac{(\ell nx)^3}{x}dx=\dfrac{1}{4}(\ell nx)^4+k}}}}\blue{\checkmark}$

$\displaystyle\sf\int e^{x^4}\cdot x^3~dx=\dfrac{1}{4}\int e^{x^4}\cdot 4x^3~dx\\\rm fac_{\!\!,}a~t=x^4\implies dt=4x^3dx\\\displaystyle\sf\int e^{x^4}\cdot 4x^3~dx=\dfrac{1}{4}\int e^t~dt=\dfrac{1}{4}e^t+k\\\huge\boxed{\boxed{\boxed{\boxed{\displaystyle\sf\int e^{x^4}\cdot x^3~dx=\dfrac{1}{4}e^{x^4}+k}}}}\blue{\checkmark}$