10) Evaluate the integral by making the given substitution.

∫sec(5x)tan(5x)dx u=5x

u=5x

du = 5dx
and the integrand becomes

1/5 sec(u) tan(u) du

That should look familiar

To evaluate the integral ∫sec(5x)tan(5x)dx using the given substitution u = 5x, follow these steps:

Step 1: Calculate du
To find du, differentiate both sides of the substitution equation with respect to x:
du/dx = d/dx (5x)
du/dx = 5

Step 2: Rearrange the integral
Substitute the expression for x from the substitution into the integral:
∫sec(u)tan(u) * (1/5) * du

Step 3: Simplify the integral
Combine the constant factor (1/5) and the integral:
(1/5)∫sec(u)tan(u) du

Step 4: Choose a suitable trigonometric substitution
To simplify the integral, we can use the substitution sec(u) = sec(u) * sec(u). Then, differentiate both sides to find du:
sec(u)tan(u) du = [sec(u) * sec(u)] du = sec^2(u) du

The integral is now simplified to:
(1/5)∫sec^2(u) du

Step 5: Evaluate the integral
The integral of sec^2(u) with respect to u is known:
∫sec^2(u) du = tan(u)

So, the integral becomes:
(1/5) * tan(u) + C

Step 6: Substitute back the original variable
Replace u with the original substitution, which is 5x:
(1/5) * tan(5x) + C

Therefore, the solution to the integral ∫sec(5x)tan(5x)dx is (1/5) * tan(5x) + C, where C is the constant of integration.