∫ cot x dx = log |sin x| + c, x ∈ I ⊂ R - {n π ; n ∈ z }
∫ cot x dx | = | |
Let f(x) | = | sin x |
Then f '(x) | = | cos x |
∴ ∫ cot x dx | = | |
= | log |f(x)| + c | |
= | log |sin x| + c |
∫ sec x dx = log |sec x + tan x| + c (or) – log |sec x – tan x| + c (or) log |tan | + c, x ∈ I ⊂ R –
∫ sec x dx | = | |
= | ||
Let f(x) = sec x + tan x. | ||
Then f '(x) = sec x . tan x + sec2 x | ||
∴ ∫ sec x dx | = | |
= | log |f(x)| + c | |
= | log |sec x + tan x| + c (or) | |
= | – log |sec x – tan x| + c (or) | |
= | log |tan | + c |
∫ cosec x dx = log |cosec x – cot x| + c (or) – log |cosec x + cot x| + c (or) log |tan | + c, x ∈ I ⊂ R – {n π ; n ∈ z}
∫ cosec x dx | = | |
= | ||
Let f(x) = cosec x – cot x | ||
Then f '(x) = – cosec x . cot x + cosec2 x | ||
∴ ∫ cosec x dx | = | |
= | log |f(x)| + c | |
= | log |cosec x – cot x| + c (or) | |
= | – log |cosec x + cot x| + c (or) | |
= | log |tan | + c |