Integral of csc^2(x): Steps and Solution with Substitution

csc^2x dx

To integrate csc^2(x) dx, we can first rewrite it in terms of sine and cosine functions:

csc^2(x) = (1/sin^2(x))

Using the identity (1 + cot^2(x)) = csc^2(x), we can rewrite the integral as follows:

∫csc^2(x) dx = ∫(1/sin^2(x)) dx

Now, let’s use a substitution to simplify the integral

To integrate csc^2(x) dx, we can first rewrite it in terms of sine and cosine functions:

csc^2(x) = (1/sin^2(x))

Using the identity (1 + cot^2(x)) = csc^2(x), we can rewrite the integral as follows:

∫csc^2(x) dx = ∫(1/sin^2(x)) dx

Now, let’s use a substitution to simplify the integral.

Let u = cos(x).

Differentiating both sides with respect to x, we get:
du = -sin(x) dx

Rearranging, we have:
-sin(x) dx = du

Substituting these values into the integral, we have:

∫(1/sin^2(x)) dx = ∫(1/(1 – u^2)) (-1 du)

Using partial fraction decomposition, we can rewrite the integral as:

-∫(1/2) (1/(1 – u)) du – ∫(1/2) (1/(1 + u)) du

Simplifying further, we have:

-1/2 ∫(1/(1 – u)) du – 1/2 ∫(1/(1 + u)) du

Using the natural logarithm property, the integral becomes:

-1/2 ln|1 – u| – 1/2 ln|1 + u| + C

Substituting back u = cos(x), we have:

-1/2 ln|1 – cos(x)| – 1/2 ln|1 + cos(x)| + C

So, the indefinite integral of csc^2(x) dx is -1/2 ln|1 – cos(x)| – 1/2 ln|1 + cos(x)| + C, where C is the constant of integration.

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