Evaluate the Integral of csc(x) Using Trigonometric Identity and Substitution

∫cscxdx

The integral of csc(x) can be evaluated by using a trigonometric identity and applying a substitution

The integral of csc(x) can be evaluated by using a trigonometric identity and applying a substitution. First, let’s rewrite csc(x) as 1/sin(x):

∫csc(x) dx = ∫(1/sin(x)) dx.

Next, we can use the identity: 1/sin(x) = csc(x)cot(x). Rewriting the integral with this identity:

∫csc(x) dx = ∫(csc(x)cot(x)) dx.

Now, we will substitute u = sin(x) so that du = cos(x) dx. This allows us to rewrite the integral in terms of u:

∫(csc(x) cot(x)) dx = ∫(csc(u) cot(x)) (1/cos(x)) du.

Simplifying the integral:

∫csc(x) dx = ∫(csc(u) /cos(x)) du.

Now, we can rewrite cot(x) /cos(x) as csc(x), using the identity cot(x) = 1/tan(x) and cos(x) = 1/sin(x):

∫csc(x) dx = ∫(csc(u) *csc(x)) du.

Combining the terms and simplifying further:

∫csc(x) dx = ∫csc(u)*csc(x) du = ∫(csc(u)*csc(x)) du = ∫(csc(u)*csc(x)) du = ∫(csc^2(u)/ sin(x)) du.

Now, let’s integrate the expression ∫(csc^2(u)/ sin(x)) du:

∫(csc^2(u)/ sin(x)) du = -cot(u)/sin(x) + C,

where C is the constant of integration.

Finally, substituting back u = sin(x):

∫csc(x) dx = -cot(sin(x))/sin(x) + C.

Therefore, the indefinite integral of csc(x) is -cot(sin(x))/sin(x) + C, where C is the constant of integration.

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