85. Another form of ∫ sec x dx
a. Verify the identity:

Question

85. Another form of ∫ sec x dx

a. Verify the identity:

sec x = cos x / (1 - sin² x)

b. Use the identity in part (a) to verify that:

∫ sec x dx = (1/2) ln |(1 + sin x)/(1 - sin x)| + C

Accepted Answer

Hi everyone, let's take a look at this practicing problem. This problem says to evaluate the integral of coasting of XDX using partial fractions. The first thing we want to do is to rewrite our integram. So, we'll rewrite cos of X as being equal to, and recall that by definition, this is 1 divided by sine of X. And we can actually rewrite this as being equal to thine of X divided by thine squared of X. And we can rewrite that denominator using our trig identity. So this is going to be equal to sin of x. Divided by the quantity of 1 minus cosine squared of x in quantity. So the integral of cosequent. Of X D X is equal to the integral of sine of x divided by the quantity of 1 minus cosine squared of X in quantity DX. So to perform this integration, we'll need to make a substitution, so we'll say U equal to cosine of X. And that means that DU is equal to minus sine of X. DX So making this substitution, our integral is going to be equal to the integral of minus 1 divided by the quantity of 1 minus u2d and quantity dU. Multiplying both the numerator and denominator by minus 1, we get that this is going to be equal to the integral of 1 divided by the quantity of U2 minus 1 and quantity DU. So that we want to do is to break the single fraction up into two separate fractions using partial fractions. And note that our denominator of a fraction here is the difference of 2 squares, so it can be factored. So we'll have 1 divided by the quantity of the quantity of u minus 1 in quantity multiplied by the quantity of u plus 1 in quantity, and this is going to be equal to a divided by the quantity of u minus 1 in quantity, plus B divided by the quantity of U plus 1. In quantity. What we want to do is multiply both sides of this equation by the quantity of u minus 1 and quantity multiplied by the quantity of u plus 1 in quantity. In doing so, we will get that 1 is equal to a multiplied by the quantity of u plus 1 in quantity, plus B multiplied by the quantity of u minus 1 in quantity. So multiplying everything out on the right hand side, we get that 1 is equal to A multiplied by U. Plus, a Plus B multiplied by U minus B. So, if we look at terms that are linear in U, we see that A plus B has to be equal to 0, since we have no linear terms on the left hand side of that equation. And if we look at the constant terms, we see that a minus B has to be equal to 1. So we take these two equations and solve for either A or B. Now, if we add these two equations together, we know that we eliminate the variable B. So when we add them together, we get that 2A is equal to 1, and divide both sides by 2, we get A is equal to 12. So we can substitute this value back into either the equations to solve for B. So, we'll use the first equation, so we'll have 1/2, plus B is equal to 0, and that means that B is equal to minus 12. So, our original equation or original integral, which is the integral. Of cosequant of XDX. is equal to The integral Of 12, divided by the quantity of U minus 1 and quantity DU. Plus the integral of minus 1/2 divided by the quantity of U plus 1 in quantity DU. Now we just need to perform the integration. In doing so, this is going to be equal to. For the first integral here, we can pull the constant factor of 1/2 out in front of the interval, so we'll have 12, and this gets multiplied by, and here we're going to be integrating 1 divided by the quantity of U minus 1 in quantity with respect to you, and recall that that gives us the natural log of the absolute value of the quantity of U minus 1 in quantity. Then we'll have minus 1/2 multiplied by interior integrating 1 divided by the quantity of u plus 1 in quantity with respect to you, and recall that's going to give us the natural log of the absolute value of the quantity of U plus 1 in quantity, then we'll have to add an integration constant of plus D to this. Now we just need to simplify this expression. So we can combine our two natural log functions used in the properties of logarithms. So this is going to be equal to 1/2 multiplied by the natural log of the absolute value of the quantity of the quantity of u minus 1 in quantity divided by the quantity of u plus 1 in quantity plus D. Now our final answer needs to be in terms of X that will substitute in cosine of X for you. In doing so, this is going to be equal to 1/2 multiplied by the natural log of the absolute value of the quantity of the quantity of cosine of X minus 1 in quantity divided by the quantity of cosine of X. Plus 1 in quantity, plus C. Now we can rewrite. The terms inside the absolute value function using our trigger identities. In doing so, this is going to be equal to 1/2 multiplied by the natural log of the absolute value of the quantity of minus 2 multiplied by the sins of the quantity of X divided by 2 in quantity. Divided by the quantity of 2 multiplied by cosine squared of the quantity of X, divided by 2 in quantity plus D. Simplifying this this expression, you see that this is going to be equal to. One half multiplied by the natural log of the absolute value of 10 squad of the quantity of X. Divided by 2 in quantity, plus D, where the minus sign inside the absolute value function has been removed since the absolute value of minus 1 is just 1. And finally, we can use the properties of logarithms to simplify this expression. In doing so, this is going to be equal to the natural log of the absolute value of tangent of the quantity of X divided by 2 in quantity plus B. This here is the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

85. Another form of ∫ sec x dx
a. Verify the identity:
sec x = cos x / (1 - sin² x)
b. Use the identity in part (a) to verify that:
∫ sec x dx = (1/2) ln |(1 + sin x)/(1 - sin x)| + C

Key Concepts

Trigonometric Identities

Integration of Trigonometric Functions

Logarithmic Integration Results

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