What is Integration of cosec 2x dx
Table of Contents
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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