Question

Determine o resultado da integral a marque a alternativa correta:

f (√x + x^x5/2) dx

A. 1/2x^-2 + 5/2x

B. 2/3x^3/2 + 2/7x^7/2 + C

C. 1/3x^-2 + 5/2x + C

D. x^3/2 + x^7/2 + C

E. x^3/2 - x^7/2 + C

Answer 1

Resposta:

$I=\sf \int\sf(\sqrt{x}+x^{\frac{5}{2}})\,dx$

• A integral da soma é igual soma das integrais.

$I=\sf \int\sf\sqrt{x}\,dx+\int\sf x^{\frac{5}{2}}\,dx$

• Passe o radical na forma de potência.

$I=\sf \int\sf x^{\frac{1}{2}}\,dx+\int\sf x^{\frac{5}{2}}\,dx$

• Aplique a regra: ∫ xⁿdx = (xⁿ⁺¹)/(n + 1) + c, n ≠ - 1.

$I=\sf \frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+\mathnormal{c}_1+\frac{x^{\frac{5}{2}+1}}{\frac{5}{2}+1}+\mathnormal{c}_2$

• A soma das constantes gera outra constantes.

$I=\sf \frac{x^{\frac{3}{2}}}{\frac{3}{2}}+\frac{x^{\frac{7}{2}}}{\frac{7}{2}}+\mathnormal{C}$

$\red{I=\sf\frac{2}{3}x^{\frac{3}{2}}+\frac{2}{7}x^{\frac{7}{2}}+\mathnormal{C}}$

Letra B