Mathematics

Solve $$\int \tan^{-1}\left(\dfrac{2x}{1-x^{2}}\right)dx$$


SOLUTION
$$\displaystyle\int \tan^{-1}\left(\dfrac{2x}{1-x^{2}}\right)dx$$
Let $$x=\tan\theta\Rightarrow \sec\theta=\sqrt{1+\tan^{2}\theta}=\sqrt{1+x^{2}}$$
$$\therefore \displaystyle\int \tan^{-1}\left(\dfrac{2\tan\theta}{1-\tan\theta}\right).\sec^{2}\theta d\theta$$
$$=\displaystyle\int \tan^{-1}(\tan 2\theta).\sec^{2}\theta\ d\theta$$
$$=\displaystyle\int 2\theta\sec^{2}\theta\ d\theta$$
$$=2\theta\tan\theta-2\displaystyle\int \tan\theta\ d\theta$$
$$=2\theta \tan\theta-2\ln |\sec\theta|+c$$
$$=2x\tan^{-1}x-2\ln |\sqrt{1+x^{2}}|+c$$

View Full Answer

Its FREE, you're just one step away


Subjective Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
Enroll Now For FREE

Realted Questions

Q1 Subjective Hard
Integrate with respect to $$x$$:
$$\sec^{3}x$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
Evaluate:
$$\displaystyle \int_{\theta =0}^{\pi}\sin \theta . d\theta$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Hard
$$\displaystyle I= \int \log \left [ x+\sqrt{x^{2}+a^{2}} \right ]dx.$$
  • A. $$\displaystyle x\log \left [ x+\sqrt{x^{2}+a^{2} } \right ]+{x^{2}+a^{2} }$$
  • B. $$\displaystyle x\log \left [ x+\sqrt{x^{2}+a^{2} } \right ]+\sqrt{x^{2}+a^{2} }$$
  • C. $$\displaystyle x\log \left [ x+\sqrt{x^{2}+a^{2} } \right ]-{x^{2}+a^{2} }$$
  • D. $$\displaystyle x\log \left [ x+\sqrt{x^{2}+a^{2} } \right ]-\sqrt{x^{2}+a^{2} }$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Hard
The value of $$\smallint {\textstyle{1 \over {x + \sqrt {x - 1} }}}$$ dx is 
  • A. $$\log \left( {x + \sqrt {x - 1} } \right) + {\sin ^{ - 1}}\sqrt {\frac{{x - 1}}{x}} + C$$
  • B. $$\log \left( {x + \sqrt {x - 1} } \right) + C$$
  • C. none of these
  • D. $$\log \left( {x + \sqrt {x - 1} } \right) - \frac{2}{3}{\tan ^{ - 1}}\left\{ {\frac{{2\sqrt {x - 1} + 1}}{{\sqrt 3 }}} \right\}C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Medium
Let $$n \space\epsilon \space N$$ & the A.M., G.M., H.M. & the root mean square of $$n$$ numbers $$2n+1, 2n+2, ...,$$ up to $$n^{th}$$ number are $$A_{n}$$, $$G_{n}$$, $$H_{n}$$ and $$R_{n}$$ respectively. 
On the basis of above information answer the following questions

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer