Question

UniCesumar Calculo

Calcule a integral indefinida abaixo, lembrando que (IMAGEM e alternativas)

Answer 1

Resposta: terceira alternativa (de cima para baixo).

Explicação passo-a-passo:

$\displaystyle\int\dfrac{x^7+x^2+1}{x^2}\ dx=\int\bigg(\dfrac{x^7}{x^2}+\dfrac{x^2}{x^2}+\dfrac{1}{x^2}\bigg)\, dx=\\\\\\ \int\big(x^5+1+x^{-2}\big)\: dx=\int x^5\:dx+\int1\:dx+\int x^{-2}\: dx=\\\\\\ \bigg(\dfrac{x^{5+1}}{5+1}+C_1\bigg)+\big(x+C_2\big)+\bigg(\dfrac{x^{-2+1}}{-2+1}+C_3\bigg)=\\\\\\ \dfrac{x^6}{6}+x+\dfrac{x^{-1}}{-1}+\underbrace{\big(C_1+C_2+C_3)}_{C}=\\\\\\ \boxed{\dfrac{x^6}{6}+x-\dfrac{1}{x}+C}$