Recognize Integral Result: We recognize that the integral of sec^2(u) with respect to u is tan(u) + C, where C is the constant of integration. This is a standard result from calculus.Find u and du/dx: Let u = 8x - 1. Then, we need to find du/dx to perform the substitution.Differentiate u: Differentiating u with respect to x, we get du/dx = 8.Find dx in terms of du: To find dx in terms of du, we rearrange the equation du = 8 dx to dx = du/8.Substitute u and dx: Now we substitute u and dx in the integral: ∫sec^2(8x-1)dx = ∫sec^2(u) * (1/8)du.Integrate sec^2(u): Using the standard integral result, we integrate sec^2(u) with respect to u to get (1/8)tan(u) + C.Substitute back for u: We substitute back for u to get the integral in terms of x: (1/8)tan(8x-1) + C.

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$\int \text{sech}^2(8x-1) \, dx$

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\int \text{sech}^2(8x-1) \, dx

Recognize Integral Result: We recognize that the integral of $\sec^2(u)$ with respect to $u$ is $\tan(u) + C$ , where $C$ is the constant of integration. This is a standard result from calculus.
Find $u$ and $\frac{du}{dx}$ : Let $u = 8x - 1$ . Then, we need to find $\frac{du}{dx}$ to perform the substitution.
Differentiate $u$ : Differentiating $u$ with respect to $x$ , we get $\frac{du}{dx} = 8$ .
Find $dx$ in terms of $du$ : To find $dx$ in terms of $du$ , we rearrange the equation $du = 8 dx$ to $dx = \frac{du}{8}$ .
Substitute $u$ and $dx$ : Now we substitute $u$ and $dx$ in the integral: $\int \sec^2(8x-1)\,dx = \int \sec^2(u) \cdot (1/8)\,du$ .
Integrate $\sec^2(u)$ : Using the standard integral result, we integrate $\sec^2(u)$ with respect to $u$ to get $(\frac{1}{8})\tan(u) + C$ .
Substitute back for u: We substitute back for u to get the integral in terms of x: $(\frac{1}{8})\tan(8x-1) + C$ .