To integrate cosec^2x, also written as ∫cosec 2 x dx, cosec squared x, cosec^2 (x), and (cosec x)^2, we start by using standard trig identities to simplify the integral. Here you will learn proof of integration of sec inverse x and cosec inverse x. Integral of square cosecant $$\int \csc^{2}x \ dx$$ maybe some of them cause a bit of conflict the resolution of this integral, but we know very well that the integrals and the derivatives are linked by a very strong friendship bond, which means that if someone derives something and the result of that something is integrated, the result will be that something with.
In this integration problem, why is ∫ sec x dx = ln sec x
∫ cscx dx = ∫ 1 u du.
X d x = ln.
How to integrate tan x. Where c is the integration constant. Lets assume u = tanx then du/dx = sec 2 x. Lets start the integration of cosec 2x.
Log a ^b = b log a log a b = b log a.therefore, − ln | cosec x + cot x | = ln 1/ | cosec x + cot x |.
= ln|cscx − cotx| + c. We can prove that the integral of cosec x to be ln | tan (x/2) | + c by using trigonometric formulas. We can write cosec x as 1/(sin x). So final expression for integration is ∫(cosec 2 x) dx = ∫(sec 2 x / tan 2 x) dx.
The indefinite integration of product of cosecant and cot functions with respect to x is equal to the sum of negative cosecant function and an integral constant.
Where c is the integration constant. Let us assume that log. Integration of cosec x formula. Here is the list of some important and most commonly asked formulas on advanced integration functions:
Cscx − cotx = 1 sinx − cosx sinx.
Integration of cosecx or cosec x. In this method of integration by substitution, any given integral is transformed into a simple form of integral by substituting the independent variable by others. Integration of cosec x in terms of log. Cosec 2 x = (1 / cos 2 x) / (sin 2 x / cos 2 x) put 1 / cos 2 x = sec 2 x and sin 2 x / cos 2 x = tan 2 x.
Integration of cosec x dx.
Or, \(\int\) cosec x = \(log |tan {x\over 2}|\) + c. X functions with respect to x is written in the following mathematical form in integral calculus. X 2 × 1 cos 2. ( x 2)) + c.
Use this property of logarithm:
Integration of cosec x dx in terms of tan. Let i = \(\int\) cosec x dx. U = cscx −cotx ⇒ du dx = −cscxcotx + csc2x, and so our integral becomes: Combine this result with the use of integration by parts, specifically, let u = x d u = d x and d v = 1 sin.
Integral of cosec x by trigonometric formulas.
[latex] ∫ cosec(x−a)cosecxdx = ∫ 1 sin(x−a)sinx dx = 1 sina ∫ sin(x−(x−a)) sinxsin(x−a) dx = 1 sina ∫ sinxcos(x−a)−cosxsin(x−a) sinxsin(x−a) dx = 1 sina ∫(cot(x−a)−cotx)dx = 1 sina(lnsin(x−a)|−ln|sinx ∣)+c = 1 sina(ln∣∣sin x−a sinx∣∣) +c ∫ cosec. We will get cosec 2 x = sec 2 x / tan 2 x; Let i = ∫ cos2(1+logtan 2x )cosec x dxput 1+logtan 2x = t⇒ tan2x 1.sec2 2x.21 dx = dt⇒ 2sin 2x cos 2x 1 dx = dt⇒ cosec xdx = dt∴ i = ∫ cos2tdt = ∫ sec2tdt= tant +c= tan(1+logtan 2x )+ c. This is the calculus part of the question complete, it now remains to show that this solution is equivalent to the given solution;
We recall the standard trig identity for cosecx, and square both sides.
