Mathematics

$$\displaystyle \int \sin^{-1}\left ( \frac{2x}{1+x^2} \right )dx$$ is equal to


ANSWER

$$\displaystyle 2(x \tan^{-1} x +\ln \left | \cos (\tan^{-1}x) \right |)+C$$


SOLUTION
$$\displaystyle I=\int \sin^{-1}\left ( \frac{2x}{1+x^{2}} \right )dx$$
Let $$x=\tan \theta $$ $$\implies \displaystyle dx=\sec ^{2}\theta d\theta $$

$$\therefore \displaystyle I=\int \sin^{-1}\left ( \frac{2\tan \theta }{1+\tan^{2}\theta} \right ) \sec^{2}\theta d\theta $$

$$=\displaystyle \int \sin^{-1}\left(\dfrac{2\frac{\sin \theta}{\cos \theta}}{1+\frac{\sin^{2}\theta}{\cos^{2}\theta}}\right)\sec^{2}\theta d\theta$$

$$=\displaystyle \int \sin^{-1}\left(\dfrac{2\frac{\sin \theta}{\cos \theta}}{\frac{\sin^{2}\theta+\cos^{2}\theta}{\cos^{2}\theta}}\right)\sec^{2}\theta d\theta$$

$$=\displaystyle \int \sin^{-1}(2\sin{\theta} \cos{\theta})\sec^{2}\theta d\theta$$

$$=\displaystyle \int \sin^{-1}(\sin{2\theta})\sec^{2}\theta d\theta$$

$$=\displaystyle 2\int \theta \sec ^{2}\theta \:d\theta $$

$$=\displaystyle 2(\theta \:\tan \:\theta +\ln \left | \cos \:\theta  \right |)+C$$

$$=\displaystyle 2(x \tan^{-1} x +\ln \left | \cos (\tan^{-1}x) \right |)+C$$

Ans: A
View Full Answer

Its FREE, you're just one step away


Single Correct 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 Medium
Calculate $$\int \frac{x}{x^{2}+1}dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Medium
$$\displaystyle\int _{ 0 }^{ \infty  }{ \dfrac { dx }{ \left( x+\sqrt { { x }^{ 2 }+1 }  \right) ^{ 3 } }  } =$$
  • A. $$\dfrac{1}{8}$$
  • B. $$-\dfrac{3}{8}$$
  • C. $$None\ of\ these$$
  • D. $$\dfrac{3}{8}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Medium
$$\displaystyle \int x^2e^{x^3}dx$$ equals
  • A. $$\dfrac {1}{3}e^{x^2}+C$$
  • B. $$\dfrac {1}{2}e^{x^3}+C$$
  • C. $$e^{-x^2}+C$$
  • D. $$\dfrac {1}{3}e^{x^3}+C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Single Correct Medium
$$\displaystyle\int^{2\pi}_{\pi}|\sin x|dx=?$$
  • A. $$0$$
  • B. $$1$$
  • C. None of these
  • D. $$2$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Medium
Evaluate $$\displaystyle \int_{1}^{3}(x^2+3x+e^{x})dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer