find-dx-x-1-x-2-and-calculate-1-3-dx-x-1-x-2-

Question Number 26397 by abdo imad last updated on 25/Dec/17

f i n d \int \frac{d x}{x \sqrt{1 + x^{2}}} a n d c a l c u l a t e \int_{1}^{3} \frac{d x}{x \sqrt{1 + x^{2}}}

Commented by abdo imad last updated on 26/Dec/17

l e t u s e t h e c h a n g e m e n t x = t a n θ

\int \frac{d x}{x \sqrt{1 + x^{2}}} = \int \frac{(1 + {t a n}^{2} θ) d θ}{t a n θ \sqrt{1 + {t a n}^{2}}} d θ = \int \frac{\sqrt{1 + {t a n}^{2} θ}}{t a n θ} d θ

= \int \frac{1}{c o s θ t a n θ} d θ = \int \frac{d θ}{s i n θ} a n d b y t h e c h a n g e m e n t t a n (θ / 2) = t

I = \int \frac{2 d t}{1 + t^{2}} {(\frac{2 t}{1 + t^{2}})}^{- 1} = \int d t / t = l n / t / = l n / t a n (\frac{a r c t a n x}{2}) / l n

\Rightarrow \int \frac{d x}{x \sqrt{1 + x^{2}}} = l n / t a n (\frac{a r c t a n x}{2}) / + k

\int_{1}^{3} \frac{d x}{x \sqrt{1 + x^{2}}} = l n / t a n (2^{- 1} a r c t a n 3) / - l n / t a n (2^{- 1} a r c t a n 1) /

== l n / t a n (\frac{a r c t a n 3}{2}) / - l n / t a n (\frac{π}{8}) / .

Answered by $@ty@m last updated on 25/Dec/17

x = \tan y

d x = \sec^{2} y d y

I = \int \frac{\sec^{2} y d y}{\tan y . \sec y}

= \int cosec y d y

= \ln ∣ cosec y - \cot y ∣ + C

= \ln ∣ cosec (\tan^{- 1} x) - \cot (\tan^{- 1} x) ∣ + C

Answered by $@ty@m last updated on 25/Dec/17

A n o t h e r m e t h o d :

L e t x = \frac{1}{t}

d x = - \frac{1}{t^{2}} d t

\int \frac{- \frac{1}{t^{2}} d t}{\frac{1}{t} \sqrt{1 + \frac{1}{t^{2}}}}

\int \frac{- \frac{1}{t^{2}} d t}{\frac{1}{t^{2}} \sqrt{t^{2} + 1}}

= \int \frac{- d t}{\sqrt{1 + t^{2}}}

= - \ln ∣ t + \sqrt{1 + t^{2}} ∣ + C

= - \ln ∣ \frac{1}{x} + \sqrt{1 + \frac{1}{x^{2}}} ∣ + C

= - \ln ∣ \frac{1 + \sqrt{1 + x^{2}}}{x} ∣ + C

= \ln ∣ \frac{x}{1 + \sqrt{1 + x^{2}}} ∣ + C

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