How to Integrate csc^2x: Step-by-Step Guide with Trigonometric Identities and Substitution

csc^2x dx

To integrate csc^2x dx, we can use a trigonometric identity and a substitution

To integrate csc^2x dx, we can use a trigonometric identity and a substitution.

First, let’s rewrite csc^2x in terms of sine and cosine. The trigonometric identity to use is:

csc^2x = 1 + cot^2x

Next, we’ll rewrite cot^2x in terms of sine and cosine using the identity:

cot^2x = cos^2x/sin^2x

Now we have:

csc^2x = 1 + cot^2x = 1 + cos^2x/sin^2x

Let’s rewrite sin^(-2)x as (1/sin^2x):

csc^2x = 1 + cos^2x/(1/sin^2x) = 1 + (cos^2x * sin^2x)

Now, let’s substitute u = sin(x), then du = cos(x) dx:

= 1 + (u^2 * du)

Integrating this expression will give us the final result:

∫ (1 + u^2) du = ∫ du + ∫ u^2 du

The integration of du is u:

= u + (1/3)u^3 + C

Substituting back the original variable, x:

= sin(x) + (1/3)sin^3(x) + C

Therefore, the integral of csc^2x dx is sin(x) + (1/3)sin^3(x) + C, where C represents the constant of integration.

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