Find-intgral-of-tan-x-by-parts-tan-x-dx-pls-help-me-i-am-a-newbie-in-calculus-

Question Number 55493 by Kunal12588 last updated on 25/Feb/19

F i n d i n t g r a l o f \tan x b y p a r t s

\int t a n x d x p l s h e l p m e (i a m a n e w b i e i n c a l c u l u s)

Commented by Kunal12588 last updated on 25/Feb/19

i a m g e t t i n g s o m e t h i n g l i k e t h i s

\int (\frac{1}{c o s x}) \times (s i n x d x)

t a k i n g u = \frac{1}{c o s x}

\Rightarrow d u = t a n x s e c x d x

t a k i n g d v = s i n x d x

\Rightarrow v = - c o s x

\int u d v = u v - \int v d u

\int t a n x d x = \frac{1}{c o s x} \times (- c o s x) - \int - c o s x \times t a n x s e c x d x

\int t a n x d x = - 1 + \int t a n x d x

b u t i n t h e b o o k : -

\int t a n x d x = - l n c o s x

Commented by maxmathsup by imad last updated on 25/Feb/19

l e t I = \int t a n x d x \Rightarrow I = \int \frac{s i n x}{c o s x} d x =_{t a n (\frac{x}{2}) = t} \int \frac{\frac{2 t}{1 + t^{2}}}{\frac{1 - t^{2}}{1 + t^{2}}} \frac{2 d t}{1 + t^{2}}

= \int \frac{4 t}{(1 + t^{2}) (1 - t^{2})} d t = \int 2 t {\frac{1}{1 + t^{2}} + \frac{1}{1 - t^{2}}} d t

= \int \frac{2 t}{t^{2} + 1} d t - \int \frac{2 t}{t^{2} - 1} d t = l n ∣ t^{2} + 1 ∣ - l n ∣ t^{2} - 1 ∣ + c

= l n ∣ \frac{1 + {t a n}^{2} (\frac{x}{2})}{1 - {t a n}^{2} (\frac{x}{2})} ∣ + c = l n ∣ \frac{1 + \frac{{s i n}^{2} (\frac{x}{2})}{{c o s}^{2} (\frac{x}{2})}}{1 - \frac{{s i n}^{2} (\frac{x}{2})}{{c o s}^{2} (\frac{x}{2})}} ∣ + c

= l n ∣ \frac{1}{c o s (x)} ∣ + c = l n ∣ c o s x ∣ + C .

Commented by tanmay.chaudhury50@gmail.com last updated on 25/Feb/19

w h a t i s t h e q u e s t i o n \int t a n x d x \to b y p a r t s

o r b y \boldsymbol p a r t i a l \boldsymbol f r a c t i o n \dots \boldsymbol p l s \boldsymbol r e c h e c k \boldsymbol t h e \boldsymbol q u e s t i o n . .

Commented by maxmathsup by imad last updated on 25/Feb/19

I = - l n ∣ c o s x ∣ + C .

Commented by maxmathsup by imad last updated on 25/Feb/19

i h a v e u s e d c h a n g e m e n t m e t h o d n o t i n t e g r a t i o n b y p a r t s \dots

Commented by Kunal12588 last updated on 26/Feb/19

s i r i t w a s n o t a q u e s t i o n I w a s t r y i n g b y m y

o w n . I w a n t t o k n o w w h e r e i s t h e m i s t a k e