Evaluate the following integrals.
∫ (1 - cosx)/(1 + cosx...

Question

Evaluate the following integrals.

∫ (1 - cosx)/(1 + cosx) dx

Accepted Answer

Hello. In this video, we are going to be evaluating the given indefinite integral. Now, we are giving the integral of 1 plus cosine of X divided by 1 minus cosine of X. Now, one way we can rewrite this integral is we can rewrite it using the half-angle identities for sine and cosine. Now, the half angle identity for sign is going to be sign of X divided by 2. And this is equal to the square root of 1 minus cosine of X divided by 2. And the half-angle identity for cosine is going to be cosine of X divided by 2, and that is equal to the square root of 1 plus cosine of X divided by 2. Now notice that we have the quantity from the denominator in the half-angle identity, and we have the quantity from our numerator of the integran in the half-angle identity of cosine. So, we can come up with substitutions for both the numerator and denominator by solving for 1 minus cosine in the sine half angle identity, and 1 plus cosine in the cosine half angle identity. By solving for 1 plus cosine of X, we will be left with 1 minus cosine of X, and that is equal to 2 squared of X divided by 2. And on the right hand side for 1 plus cosine of X, solving for that quantity leaves us with 1 plus cosine of X equal to 2 cosine squared of X divided by 2. Now, let's go ahead and apply these substitutions to the integrand. If we apply these substitutions, we can rewrite the integral to be 2. Cosine squared. Of X divided by 2. Divided by 2 squared of X divided by 2. Here, we can simplify 2 from both the numerator and denominator, leaving us with cosine squared divided by sine squared, and that is the trigonometric identity for cotangent. So we can rewrite the integral to be the integral of cotangent squared of x divided by 2D X. Now, let's go ahead and simplify this integral by using a U substitution. If we allow U to equal to X divided by 2, Then DU is equal to 1/2 DX, and solving for DX in this case by multiplying both sides of the equation by 2, is going to leave us with 2 DU is equal to DX. So we can rewrite the integral to be 2 multiplied by the integral of cotangent squared of UDU. Here we can further simplify cotangent by using its Pythagorean identity. We can rewrite cotangent squared of x as coskin squared of u minus 1, so we can rewrite the integral as 2 multiplied by the integral of cos squared of u minus 1 D U. And by taking the antiderivative of each of these terms, we will have two multiplied by negative cotangent of U minus 1, or minus, sorry. And because we were solving for a definite indefinite integral, we will add a general constant C. Now, if we were to distribute 2 into that group, we will be left with negative 2 cotangent of U minus 2 U plus C. And now what we need to do is we need to backs substitute U into terms of X. Now earlier we stated that U is equal to X divided by 2. And so we will have -2 cotangent of X divided by 2 minus 2 multiplied by X divided by 2, which is just X plus C, and this is going to be the general solution to the given indefinite integral. So, I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Evaluate the following integrals.
∫ (1 - cosx)/(1 + cosx) dx

Key Concepts

Integration

Trigonometric Identities

Substitution Method

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