(13) $\\int \\frac{7x^{6}+3}{3x+x^{7}}dx$

Explanation:

Step1: Let (u = 3x + x^{7})

Differentiate (u) with respect to (x): (du=(3 + 7x^{6})dx)

Step2: Substitute (u) and (du) into the integral

The integral (\int\frac{7x^{6}+3}{3x + x^{7}}dx=\int\frac{du}{u})

Step3: Integrate (\frac{du}{u})

We know that (\int\frac{du}{u}=\ln|u|+C) (where (C) is the constant of integration)

Step4: Substitute back (u = 3x + x^{7})

We get (\ln|3x + x^{7}|+C)

Answer:

(\ln|3x + x^{7}|+C)