Evaluate ∫ sec θ dθ by:
a. Multiplying by (sec θ + tan θ...

Question

Evaluate ∫ sec θ dθ by:

a. Multiplying by (sec θ + tan θ) / (sec θ + tan θ) and then using a u-substitution.

Accepted Answer

Welcome back everyone. Evaluate the integral of cosequent x dx by multiplying by cosequent x minus cotangent x divided by cosequent x minus cotangent x and using a substitution. So what we're going to do is rewrite the integral as cosequent x multiplied by cosequent x minus cotangent x. And this is divided by Cosequent x minus cotangent x followed by d x. We're going to introduce substitution. Let's suppose that u is equal to cosequent x minus cotangent x. Let's remember that we want to find the derivative of e with respect to x. The derivative of cosequent x is negative cosequent x cotangent x, and we're subtracting the derivative of cotangent x, which is negative cosequent squared x. That be co sequin squared x minus cosequent x cotangent x. And essentially what we can do is factor out negative. Cose and a. So that we will have. Negative cosequent x. Plus cotangent x. So instead let's just factor out cosequent x, which will give us cosequent x multiplied by cosequent x minus cotangent x. So now our signs are consistent, right? And what we can notice is that d u is equal to cosequent x multiplied by cosequent x minus cotangent x d x. This is exactly what we have in the numerator, so we can rewrite the integral as the integral of d u divided by u, which is an elementary integral. Its value is ln of the absolute value of u plus a constant of integration c. And therefore the final answer is ln of the absolute value of cosequent x. Minus cotangent x Plus a constant of integration c. Thank you for watching.

Evaluate ∫ sec θ dθ by:a. Multiplying by (sec θ + tan θ) / (sec θ + tan θ) and then using a u-substitution.

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Evaluate ∫ sec θ dθ by:
a. Multiplying by (sec θ + tan θ) / (sec θ + tan θ) and then using a u-substitution.