Evaluate the integral. int(8e^((8sin 7x)))/(sec 7x)dx

Question

Evaluate the integral.  int(8e^((8sin 7x)))/(sec 7x)dx

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Simplify the Integral: Let's start by simplifying the integral. We know that sec(7x) is the reciprocal of cos(7x), so we can rewrite the integral as: ∫(8e^(8sin(7x)) * cos(7x))dx This looks like a good candidate for a substitution because the derivative of sin(7x) is related to cos(7x).Substitution with u: Let u = 8sin(7x). Then, we need to find du in terms of dx. We know that the derivative of sin(7x) with respect to x is 7cos(7x), so: du/dx = 8 * 7cos(7x) du = 56cos(7x)dx Now we need to express cos(7x)dx in terms of du.Express cos(7x)dx in terms of du: We have du = 56cos(7x)dx, so we can solve for cos(7x)dx: cos(7x)dx = du/56 Now we can substitute u and du/56 into the integral: ∫(8e^u * (du/56)) This simplifies to: (1/7)∫e^u duSubstitute u back in: The integral of e^u with respect to u is simply e^u. So we have: (1/7)e^u + C Now we need to substitute back in for u to get the integral in terms of x.Final Answer: We originally let u = 8sin(7x), so we substitute back in to get: (1/7)e^(8sin(7x)) + C This is our final answer.

Evaluate the integral. $\newline$ $\int \frac{8e^{(8\sin 7x)}}{\sec 7x}\,dx$

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Evaluate the integral.\newline∫8e(8sin⁡7x)sec⁡7x dx\int \frac{8e^{(8\sin 7x)}}{\sec 7x}\,dx∫sec7x8e(8sin7x)​dx

Evaluate the integral. $\newline$ $\int \frac{8e^{(8\sin 7x)}}{\sec 7x}\,dx$