Final answer:
To find the indefinite integral of arcsec(8x) dx, we use a substitution technique. By letting u = 8x and finding du, we can convert the integral to one involving arcsec(u). Afterward, we use the formula for integrating arcsec(u) to calculate the integral and obtain the final result.
Step-by-step explanation:
To find the indefinite integral of arcsec(8x) dx, we can use a substitution. Let u = 8x. Then, du = 8 dx. Rearranging this equation, we have dx = du/8. Now, we can rewrite the integral as ∫ arcsec(u) (du/8).
To integrate arcsec(u), we can use the formula:
∫ arcsec(u) du = u arcsec(u) + sqrt(u^2 - 1) + C,
where C is the constant of integration.
So, substituting back u = 8x, we get:
∫ arcsec(8x) dx = (8x) arcsec(8x) + sqrt((8x)^2 - 1)/8 + C
Therefore, the indefinite integral of arcsec(8x) dx is (8x) arcsec(8x) + sqrt((8x)^2 - 1)/8 + C.