Question

evaluate the indefinite integral.\\(\int \sqrt7{x}\\ dx =\\)

Explanation:

Step1: Rewrite radical as power

$\sqrt[7]{x} = x^{\frac{1}{7}}$

Step2: Apply power rule for integration

The power rule is $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ where $n
eq -1$. Here $n=\frac{1}{7}$, so:
$\int x^{\frac{1}{7}} dx = \frac{x^{\frac{1}{7}+1}}{\frac{1}{7}+1} + C$

Step3: Simplify exponent and denominator

$\frac{1}{7}+1 = \frac{8}{7}$, so:
$\frac{x^{\frac{8}{7}}}{\frac{8}{7}} + C = \frac{7}{8}x^{\frac{8}{7}} + C$

Step4: Rewrite back to radical form (optional)

$\frac{7}{8}x^{\frac{8}{7}} = \frac{7}{8}x\sqrt[7]{x}$, so:
$\frac{7}{8}x\sqrt[7]{x} + C$

Answer:

$\frac{7}{8}x^{\frac{8}{7}} + C$ (or equivalently $\frac{7}{8}x\sqrt[7]{x} + C$)