Solving the Integral ∫cot(u)du using Integration by Substitution and Trigonometric Identities

∫cot u du

To solve the integral ∫cot(u)du, we can use integration by substitution or a trigonometric identity

To solve the integral ∫cot(u)du, we can use integration by substitution or a trigonometric identity.

Using Integration by Substitution:
Let’s start by letting w = sin(u). Then, differentiate both sides to find dw.

dw/dx = cos(u) du
du = (dw)/(cos(u))

Now, substitute the value of du into the integral:

∫cot(u)du = ∫cot(u) (dw/cos(u))

Cancel out the cos(u) term:

∫cot(u)du = ∫cot(u)dw

Using the identity cot(u) = cos(u)/sin(u):

∫cot(u)du = ∫(cos(u)/sin(u))dw

Now, we need to convert the expression in terms of w and differentiate the denominator:

∫cot(u)du = ∫(cos(u)/sin(u))dw
= ∫(1/sin(u))cos(u)dw

Using the identity cos(u)du = dw:

∫cot(u)du = ∫(1/sin(u))cos(u)dw
= ∫(1/w)dw
= ln|w| + C

Substituting back, we get:

∫cot(u)du = ln|sin(u)| + C

Therefore, the integral ∫cot(u)du is equal to ln|sin(u)| + C, where C is the constant of integration.

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