A Comprehensive Guide on Finding the Antiderivative of csc^2x: Integration by Substitution Technique

Antiderivative csc^2x

To find the antiderivative of csc^2x, we can use a technique called integration by substitution

To find the antiderivative of csc^2x, we can use a technique called integration by substitution.

Let’s start by recalling the identity: csc^2x = 1 + cot^2x.

Now, let’s rewrite the expression of the antiderivative: ∫csc^2x dx.

We can rewrite csc^2x as (1 + cot^2x).

So, our integral becomes: ∫(1 + cot^2x) dx.

Now, we can split the integral into two parts: ∫1 dx + ∫cot^2x dx.

The integral of 1 with respect to x is simply x + C1, where C1 is the constant of integration.

Now, we are left with the second integral: ∫cot^2x dx.

To solve this integral, we can use the trigonometric identity: cot^2x = csc^2x – 1.

So, our integral becomes: ∫(csc^2x – 1) dx.

Distributing the integral: ∫csc^2x dx – ∫1 dx.

The first integral is what we initially wanted to solve, which is the antiderivative of csc^2x. The second integral is just x.

Therefore, the second integral is x and the first integral, according to our solved identity, is -cotx.

So, the final answer is: x – cotx + C2, where C2 is the constant of integration.

In summary, the antiderivative of csc^2x is x – cotx + C, where C is the constant of integration.

More Answers: