Mastering Integral Calculus | How to Solve the Integral of cot(x) with Step-by-Step Substitution

∫ cot x dx

To solve the integral of cot(x) with respect to x, you can use a technique called integration by substitution

To solve the integral of cot(x) with respect to x, you can use a technique called integration by substitution.

First, let’s rewrite cot(x) in terms of sine and cosine. Recall that cot(x) is the reciprocal of tan(x), which is equal to cos(x) divided by sin(x). Therefore, cot(x) can be expressed as cos(x) / sin(x).

Now, let’s perform a substitution by letting u = sin(x). This means that du = cos(x) dx, which we can rearrange to obtain dx = du / cos(x).

Using these substitutions, we can rewrite the integral as follows:
∫ cot(x) dx = ∫ (cos(x) / sin(x)) dx.

Substituting the expressions for dx and cot(x) in terms of u gives us:
∫ (cos(x) / sin(x)) dx = ∫ (cos(x) / sin(x)) (du / cos(x)).

We can simplify this expression by canceling out the cos(x) terms:
∫ (cos(x) / sin(x)) (du / cos(x)) = ∫ (1 / sin(x)) du.

Now, the integral has been transformed into a more manageable form.

Next, we can transform the expression 1 / sin(x) using the Pythagorean identity:
1 / sin(x) = csc(x).

Substituting this expression back into the integral, we get:
∫ csc(x) du.

At this point, we recognize that the integral of csc(x) with respect to x is the natural logarithm of the absolute value of the cosecant of x, plus a constant. So, our final solution is:
∫ cot(x) dx = ∫ csc(x) du = ln|csc(x)| + C,

where C represents the constant of integration.

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