Calculate the integral of (7x^5) dx.

To calculate the integral of (7x^5) with respect to (x), we use the power rule for integration. The power rule states that the integral of (x^n) is given by (\frac{x^{n+1}}{n+1}), plus a constant of integration (C), where (n) is any real number except (-1).

In this case, we have the function (7x^5). We can factor out the constant (7) before applying the power rule:

[

\int 7x^5 , dx = 7 \int x^5 , dx

]

Now applying the power rule, we increase the exponent by 1 (from 5 to 6) and divide by the new exponent (6):

[

\int x^5 , dx = \frac{x^{6}}{6}

]

So now substituting back, we get:

[

7 \int x^5 , dx = 7 \cdot \frac{x^{6}}{6} = \frac{7}{6}x^{6}

]

Finally, we add the constant of integration (C