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Integration of cosec 2x

Integration of cosec 2x dx

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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