The integration of cosec x is: ∫cosec x dx = ln|tan(x/2)| + C. This is equivalent to –ln|cosec x + cot x| + C or ln|cosec x – cot x| + C. Here, ln denotes the natural logarithm and C is the constant of integration. This result is part of the standard integral formulae for Class 12 Maths (NCERT).
∫cosec x dx = ln|tan(x/2)| + C (standard result).
Equivalent form: –ln|cosec x + cot x| + C.
Derived using multiply-divide by (cosec x – cot x) and substitution.
∫cosec²x dx = –cot x + C (different formula).
∫cosec x cot x dx = –cosec x + C.
All trigonometric integrals are standard formulae in NCERT Class 12 Maths.
Standard Result: ∫cosec x dx = ln|tan(x/2)| + C
Equivalent Forms: • ∫cosec x dx = –ln|cosec x + cot x| + C • ∫cosec x dx = ln|cosec x – cot x| + C (All three forms are equivalent; they differ by a constant)
Derivation (Multiply and Divide Method): ∫cosec x dx = ∫cosec x × (cosec x – cot x)/(cosec x – cot x) dx = ∫(cosec²x – cosec x cot x)/(cosec x – cot x) dx
Let t = cosec x – cot x dt = (–cosec x cot x + cosec²x) dx = (cosec²x – cosec x cot x) dx
So: ∫dt/t = ln|t| + C = ln|cosec x – cot x| + C
Alt Form — Using Half Angle: cosec x – cot x = tan(x/2) ∴ ∫cosec x dx = ln|tan(x/2)| + C
Key Related Integrals:
| Integral | Result |
|---|---|
| ∫sin x dx | –cos x + C |
| ∫cos x dx | sin x + C |
| ∫tan x dx | ln |
| ∫cot x dx | ln |
| ∫sec x dx | ln |
| ∫cosec x dx | ln |
| ∫cosec²x dx | –cot x + C |
| ∫cosec x cot x dx | –cosec x + C |
Solved Example: Find ∫cosec(2x) dx • Use substitution: let u = 2x, du = 2dx • ∫cosec(2x) dx = (1/2)∫cosec u du • = (1/2) ln|tan(u/2)| + C • = (1/2) ln|tan(x)| + C
The integration of cosec x is: ∫cosec x dx = ln|tan(x/2)| + C. Equivalent forms are –ln|cosec x + cot x| + C or ln|cosec x – cot x| + C. The derivation uses the multiply-divide trick with (cosec x – cot x) and substitution. Note: ∫cosec²x dx = –cot x + C is a different result.
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