Integration of cosecx: Formula, Proof | Cosecx Integration

The integration of cosec x is ln|cosec x-cot x|. Here ln stands for the natural logarithm, that is, ln x = logex. In this post, we will learn how to integrate cosec x.

Integration of Cosecx Formula

Cosecx integration formula: The integration formula of cosec x is given below.

∫cosecx dx = ln|cosecx-cotx|+C

where C is an integral constant.

Integration of cosecx

Integration of Cosecx Proof

We will prove that ∫cosecx dx = ln|cosecx-cotx|+C. Note that

∫cosecx dx = ∫cosecx × 1 dx

= ∫cosecx $\times \dfrac{\text{cosec} x -\cot x}{\text{cosec} x – \cot x}$ dx

= ∫ $\dfrac{\text{cosec}^2 x – \text{cosec} x\cot x}{\text{cosec} x – \cot x}$ dx …(I)

Put cosecx – cotx = t.

Differentiating both sides, we have

(-cosec x cot x+cosec2x)dx = dt.

Thus, we have from (I) that

∫cosecx dx = ∫ $\dfrac{dt}{t}$

= ln|t|+C where C is an integral constant, and ln stands for the natural logarithm which is loge.

= ln|cosecx – cotx|+C as t=cosecx – cotx.

So the integration of cosecx is ln|cosecx – cotx|+C where C is an integral constant. This is derived by the substitution method of integration.

Video Solution on Integration of cosecx:


Integration of tan x

Integration of cot x

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Q1: What is the integration of cosecx?

Answer: The integration of cosecx is equal to ln|cosecx – cotx|+C.

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