How To Integrate Cot(X) Using The Method Of Substitution

∫cot(x)dx

-ln|csc(x)| +C

To integrate ∫cot(x)dx, we can use the method of substitution. Let’s consider the following substitution:

u = sin(x)

Then, we can express cot(x) in terms of u:

cot(x) = cos(x) / sin(x) = √(1 – sin^2(x)) / sin(x) = √(1 – u^2) / u

We can also express dx in terms of u:

du / dx = cos(x)

dx = du / cos(x)

Substituting these expressions into the integral, we obtain:

∫cot(x)dx = ∫(√(1 – u^2) / u) (du / cos(x))

Simplifying this expression, we get:

∫cot(x)dx = ∫(√(1 – u^2) / u) sec(x) du

Now, let’s recall a trigonometric identity:

sec(x) = 1 / cos(x)

Substituting this into the integral, we obtain:

∫cot(x)dx = ∫(√(1 – u^2) / u) (1 / cos(x)) cos(x) du

Simplifying this expression, we get:

∫cot(x)dx = ∫√(1 – u^2) du

This integral can be evaluated using the substitution u = sin(t):

u = sin(t)

du = cos(t) dt

Substituting these expressions into the integral, we obtain:

∫cot(x)dx = ∫√(1 – u^2) du = ∫cos(t) cos(t) dt

∫cot(x)dx = ∫cos^2(t) dt

Using the trigonometric identity:

cos^2(t) = (1 + cos(2t)) / 2

We can express the integral as:

∫cot(x)dx = (1/2) ∫(1 + cos(2t)) dt

∫cot(x)dx = (1/2)(t + (1/2) sin(2t)) + C

Substituting back u = sin(x), we obtain:

∫cot(x)dx = (1/2)(arcsin(u) + (1/2) sin(2arcsin(u))) + C

Therefore, the antiderivative of cot(x) is:

∫cot(x)dx = (1/2)(arcsin(sin(x)) + (1/2) sin(2arcsin(sin(x)))) + C

Simplified as:

∫cot(x)dx = (1/2)(x + sin(x) cos(x)) + C

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