Evaluate the integral. HINT [See Example 2 and 4.] (Remember to use ln |u| where appropriate.)

(1/v7 + 5/v) dv

the 1/v7 is confusing.

please clarify .

its supposed to be 1/v^7

then integral is

(-1/6)v^-6 + 5ln|v| + c

To evaluate the integral ∫(1/v^7 + 5/v) dv, we can split it into two separate integrals since they have different terms.

The first term is ∫(1/v^7) dv.

To integrate 1/v^7, we can use the power rule for integration. The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C.

Applying the power rule, we have ∫(1/v^7) dv = (-1/(7-1)) * v^(-1/7) + C = (-1/6) * v^(-1/7) + C1.

The second term is ∫(5/v) dv.

To integrate 5/v, we can rewrite it as 5v^(-1). Again, applying the power rule, we have ∫(5/v) dv = 5 * v^0/0 + C = 5 * ln(|v|) + C2.

Finally, we can combine the two integrals to find the overall integral.

∫(1/v^7 + 5/v) dv = ∫(1/v^7) dv + ∫(5/v) dv = (-1/6) * v^(-1/7) + 5 * ln(|v|) + C1 + C2.

Since C1 and C2 are arbitrary constants, we can combine them as a single constant C.

Therefore, the final answer is (-1/6) * v^(-1/7) + 5 * ln(|v|) + C.