Integrating cscx – A Step-by-Step Guide with Integration by Substitution

∫(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.

More Answers: