Evaluate the indefinite integral.

(e^7x)/(e^14x+16)dx

Use the substitution:

u=e^(7x)
then
du = 7e^(7x)dx
and the integral
I=∫e^(7x)/(e^(14x)+16)dx
=(1/7)∫du/(u²+16)
which is a standard form that for arctan
=(1/7)(1/4)tan-1(u/4)
Back-substitute u=e^(7x) into the expression to get the answer in terms of x.

To evaluate the indefinite integral ∫(e^7x) / (e^14x+16) dx, we can start by using a substitution method. Let's make the substitution u = e^7x so that du/dx = 7e^7x, which implies dx = (1/7e^7x) du.

Now we can rewrite the integral using the new substitution:
∫[(e^7x) / (e^14x+16)] dx = ∫[(u) / (e^14x+16)] *(1/7e^7x) du

Next, we need to express e^14x in terms of u. We can rewrite it as e^(2*7x), which is (e^7x)^2, so e^14x = (u)^2.

Replacing e^14x in the integral expression:
∫[(u) / ((u)^2+16)] * (1/7e^7x) du

Now we can simplify the expression further:
∫[(u) / (u^2+16)] * (1/7e^7x) du = (1/7) * ∫[(u) / (u^2+16)] * (1/e^7x) du

To proceed, let's split the fraction using partial fractions. We write (u) / (u^2+16) as:
(u) / (u^2+16) = A/(u+4) + B/(u-4)

To find the values of A and B, we can use a common denominator:
(u) = A(u-4) + B(u+4)
Let's solve for A and B.

Setting u = 4, we get:
4 = A(4-4) + B(4+4)
4 = 8B
B = 1/2

Setting u = -4, we get:
-4 = A(-4-4) + B(-4+4)
-4 = -8A
A = 1/2

Now we can rewrite the integral:
(1/7) * ∫[(u) / (u^2+16)] * (1/e^7x) du = (1/7) * ∫[(1/2) / (u+4)] + [(1/2) / (u-4)] * (1/e^7x) du

Next, we can simplify further:
(1/7) * ∫[(1/2) / (u+4)] + [(1/2) / (u-4)] * (1/e^7x) du = (1/14) * ∫[(1 / (u+4)) + (1 / (u-4))] * (1/e^7x) du

The integral ∫[(1 / (u+4)) + (1 / (u-4))] * (1/e^7x) du is straightforward to evaluate:
∫[(1 / (u+4)) + (1 / (u-4))] * (1/e^7x) du = (1/14) * [ln|u+4| + ln|u-4|] * (1/e^7x) + C

Finally, we substitute back in the original variable:
(1/14) * [ln|u+4| + ln|u-4|] * (1/e^7x) + C = (1/14e^7x) * [ln|e^7x+4| + ln|e^7x-4|] + C

And that is the final result.