Question

In order to evaluate 3 sec θ dθ , multiply the integrand by sec θ + tan θ sec θ + tan θ . 3 sec θ dθ = 3 sec(θ) sec θ + tan θ sec θ + tan θ dθ = 3 sec2 θ + sec θ tan θ sec θ + tan θ dθ Since d dθ (sec θ + tan θ) . , we can evaluate sec θ dθ with a u-substitution of u = . Apply this u-substitution and write the answer in terms of θ. (Remember to use absolute values where appropriate.)

Answer 1

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Explanation:

∫3 sec θ dθ

Multiply the denominator and numerator by sec θ + tan θ

∫3 secθ × (secθ + tanθ) / (secθ + tanθ) dθ

3∫(sec²θ + secθtanθ) / (secθ + tanθ) dθ

Let U = secθ + tanθ

dU / dθ = secθtanθ + sec²θ

dθ = dU / secθtanθ + sec²θ

Then,

3∫(sec²θ + secθtanθ) / (secθ + tanθ) dθ

3∫(sec²θ + secθtanθ) / (secθ + tanθ) × dU / secθtanθ + sec²θ

3∫1 / U × dU

3∫dU / U

3 In|U| + C

3• In|secθ + tanθ| + C