Calcule a seguinte integral indefinida ∫ √[3]{7 + x^5} x^4 dx

Queremos calcular

$$\int (7+x^5)^{1/3}\,x^4\,dx.$$

Passo 1: Substituição Tome

$$u = 7+x^5 \quad\Rightarrow\quad d\nu = 5x^4\,dx \quad\Rightarrow\quad x^4\,dx = \frac{1}{5}d\nu.$$

Passo 2: Reescrever a integral em $\nu$

$$\int (7+x^5)^{1/3}x^4\,dx = \int \nu^{1/3}\cdot \frac{1}{5}d\nu = \frac{1}{5}\int \nu^{1/3}d\nu.$$

Passo 3: Integrar Usando a regra $\int \nu^a d\nu = \frac{\nu^{a+1}}{a+1}+C$ (para $a\neq -1$), com $a=\frac{1}{3}$:

$$\frac{1}{5}\int \nu^{1/3}d\nu = \frac{1}{5}\cdot \frac{\nu^{4/3}}{4/3} + C = \frac{1}{5}\cdot \frac{3}{4}\nu^{4/3}+C = \frac{3}{20}\nu^{4/3}+C.$$

Passo 4: Voltar para $x$

$$\frac{3}{20}\nu^{4/3}+C = \frac{3}{20}(7+x^5)^{4/3}+C.$$