Integration of cosec 2x

What is Integration of cosec 2x dx

We can rewrite Integration of cosec 2x as ∫cosec²x dx, or Integral of cosec square x or cosec^2x or (cosec x)² . We will use trigonometric identities to find the Integral of cosec^2x. As you know that cosec x = 1/sinx so cosec² x = 1/ sin² x. Now We will divide both numerator and denominator by cos²x.

∫cosec²x dx = ?

• cosec x = 1/sinx
• cosec² x = 1/ sin² x. Now We will divide both numerator and denominator by cos²x.
• So our next step will become cosec² x = (1 / cos²x) / (sin² x / cos²x)

Find the Integral of cosec 2x dx

Now we can see two more trigonometric identities 1 / cos²x = sec² x and anther one is sin² x / cos²x = tan²x. We will substitute these identities into previous expression cosec² x = (1 / cos²x) / (sin² x / cos²x). After that we will find the Integration of cosec 2x or cosec square x.

• cosec² x = (1 / cos²x) / (sin² x / cos²x) Put 1 / cos²x = sec² x and sin² x / cos²x = tan²x.
• We will get cosec² x = sec² x / tan²x

So final expression for integration is ∫(cosec² x) dx = ∫(sec² x / tan²x) dx. Lets start the Integration of cosec 2x. Lets assume u = tanx then du/dx = sec²x. New we can rearrange it as du = sec² x dx because same term is available in our integration expression.

∫(cosec² x) dx

• ∫(cosec² x) dx = ∫(sec² x / tan²x) dx
• Lets assume u = tanx
• So du/dx = sec²x
• Rewrite as du = sec² x dx

After substituting these u = tanx and sec² x dx = du. Now we will get the complete expression in terms of u and du. Integrate this with respect to U so we can get Integration of cosec 2x

• ∫(cosec² x) dx = ∫(sec² x / tan²x) dx = ∫ (1/ u² ) du. Now we got the complete expression in terms of u and du
• ∫ (1/ u² ) du can be written as ∫ u^-2 du. Now will integrate this with respect to U.
• ∫ (1/ u² ) du = {1 / (-1)} * (u^-1) + C. Where C is a integral constant

Previous expression ∫ (1/ u² ) du = {1 / (-1)} * (u^-1) + C can be written as ∫ (1/ u² ) du = – (u^-1 ) + C = – (1 / u) + C

• Finally we got ∫ (1/ u² ) du = – (1 / u) + C. Now we can replace u = tanx
• ∫ (1/ u² ) du = – (1 / tanx) + C

Now we can see one more trigonometric identity tanx = 1/ cotx so cotx = 1 / tanx. Substitute this 1 / tanx = cotx in previous expression

• ∫ (1/ u² ) du = – (1 / tanx) + C
• ∫ (1/ u² ) du = – cotx + C

So finally we can say that ∫(cosec² x) dx = – cotx + C . Please follow this link of Explanation of Mathematics formula – Click

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