∫(cscx)dx
To integrate the function ∫(cscx)dx, we can use a technique called integration by substitution
To integrate the function ∫(cscx)dx, we can use a technique called integration by substitution.
Let’s start by rewriting the integral using the reciprocal identity for the csc function:
∫(cscx)dx = ∫(1/sinx)dx
Now, let’s make a substitution by letting u = sinx.
Differentiating both sides with respect to x, we get du/dx = cosx.
Solving for dx, we have dx = du/cosx.
Substituting these values into the integral, we get:
∫(1/sinx)dx = ∫(1/u)(du/cosx)
Now, we need to replace dx and sinx with the appropriate values in terms of u.
When x = 0, u = sin0 = 0, and when x = π/2, u = sin(π/2) = 1.
So, our new limits of integration become u = 0 to u = 1.
Substituting these new values and the dx substitution into the integral, we have:
∫(1/sinx)dx = ∫(1/u)(du/cosx)
= ∫(du)/(u*cosx)
Now, we can integrate with respect to u:
∫(1/u)(du/cosx) = ∫(du)/(u*cosx)
= (1/cosx)∫(du/u)
= (1/cosx)ln|u| + C
Remembering that u = sinx, we can substitute it back into the integral expression:
(1/cosx)ln|sinx| + C
Therefore, the integral of cscx is (1/cosx)ln|sinx| + C, where C is the constant of integration.
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