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Answer:
[tex]\int {7 \sec(\theta) } \, d\theta = 7\ln(\sec(\theta) + \tan(\theta)) + c[/tex]
Step-by-step explanation:
The question is not properly formatted. However, the integral of [tex]\int {7 \sec(\theta) } \, d\theta[/tex] is as follows:
[tex]\int {7 \sec(\theta) } \, d\theta[/tex]
Remove constant 7 out of the integrand
[tex]\int {7 \sec(\theta) } \, d\theta = 7\int {\sec(\theta) } \, d\theta[/tex]
Multiply by 1
[tex]\int {7 \sec(\theta) } \, d\theta = 7\int {\sec(\theta) * 1} \, d\theta[/tex]
Express 1 as: [tex]\frac{\sec(\theta) + \tan(\theta) }{\sec(\theta) + \tan(\theta)}[/tex]
[tex]\int {7 \sec(\theta) } \, d\theta = 7\int {\sec(\theta) * \frac{\sec(\theta) + \tan(\theta) }{\sec(\theta) + \tan(\theta)}} \, d\theta[/tex]
Expand
[tex]\int {7 \sec(\theta) } \, d\theta = 7\int {\frac{\sec^2(\theta) + \sec(\theta)\tan(\theta) }{\sec(\theta) + \tan(\theta)}} \, d\theta[/tex]
Let
[tex]u = \sec(\theta) + \tan(\theta)[/tex]
Differentiate
[tex]\frac{du}{d\theta} = \sec(\theta)\tan(\theta) + sec^2(\theta)[/tex]
Make [tex]d\theta[/tex] the subject
[tex]d\theta = \frac{du}{\sec(\theta)\tan(\theta) + sec^2(\theta)}[/tex]
So, we have:
[tex]\int {7 \sec(\theta) } \, d\theta = 7\int {\frac{\sec^2(\theta) + \sec(\theta)\tan(\theta) }{u}} \,* \frac{du}{\sec(\theta)\tan(\theta) + sec^2(\theta)}[/tex]
Cancel out [tex]\sec(\theta)\tan(\theta) + sec^2(\theta)[/tex]
[tex]\int {7 \sec(\theta) } \, d\theta = 7\int {\frac{1}{u}} \,du}}[/tex]
Integrate
[tex]\int {7 \sec(\theta) } \, d\theta = 7\ln(u) + c[/tex]
Recall that: [tex]u = \sec(\theta) + \tan(\theta)[/tex]
[tex]\int {7 \sec(\theta) } \, d\theta = 7\ln(\sec(\theta) + \tan(\theta)) + c[/tex]