Mathematics

$$\displaystyle \int { \cfrac { x\tan ^{ -1 }{ x }  }{ { \left( 1+x^{ 2 } \right)  }^{ 3/2 } }  } dx=$$


ANSWER

$$\cfrac { x-\tan ^{ -1 }{ x } }{ { \sqrt { \left( 1+x^{ 2 } \right) } }^{ } } +c$$


SOLUTION
Let $$I = \displaystyle \int \dfrac{x \tan^{-1} x}{(1 + x^2)^{3/2}} dx $$         let $$x = \tan \theta$$
                                                         $$dx = \sec^2 \theta d \theta$$

$$I = \displaystyle \int \dfrac{\tan \theta . \theta . \sec^2 \theta}{\sec^3 \theta} d \theta = \int \dfrac{\theta \tan \theta }{\sec \theta } d \theta$$

$$I = \displaystyle \int \theta. \sin \theta \, d \theta = - \theta \cos \theta + \int \cos \theta $$

$$= - \theta \cos \theta + \sin \theta$$

$$= \dfrac{-\tan^{-1} x}{\sqrt{1 + x^2}} + \dfrac{x}{\sqrt{1 + x^2}} = \dfrac{x - \tan^{-1} x}{\sqrt{1 + x^2}} + C$$
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 84
Enroll Now For FREE

Realted Questions

Q1 Single Correct Hard
If $$\displaystyle\int { { x }^{ 13/2 }.{ \left( 1+{ x }^{ 5/2 } \right)  }^{ 1/2 }dx } =P{ \left( 1+{ x }^{ 5/2 } \right)  }^{ 7/2 }+Q{ \left( 1+{ x }^{ 5/2 } \right)  }^{ 5/2 }+R{ \left( 1+{ x }^{ 5/2 } \right)  }^{ 3/2 }+C$$, then $$P,\ Q$$ and $$R$$ are
  • A. $$P=\frac { 4 }{ 35 } ,\ Q=\frac { 8 }{ 25 } ,\ R=\frac { 4 }{ 15 } $$
  • B. $$P=-\frac { 4 }{ 35 } ,\ Q=-\frac { 8 }{ 25 } ,\ R=\frac { 4 }{ 15 } $$
  • C. $$P=\frac { 4 }{ 35 } ,\ Q=-\frac { 8 }{ 25 } ,\ R=-\frac { 4 }{ 15 } $$
  • D. $$P=\frac { 4 }{ 35 } ,\ Q=-\frac { 8 }{ 25 } ,\ R=\frac { 4 }{ 15 } $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
Evaluate the following definite integral.

$$\displaystyle \int _{2}^3 \dfrac {x}{x^2+1}dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Evaluate $$\displaystyle \int_{0}^{\pi/2} cos \,x \,e^{sin \,x} \,dx$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Evaluate the integral $$\displaystyle\int_{-4}^{4}|x+2|\ dx$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Medium
Let $$\displaystyle I_{1}=\int_{0}^{1}(1-x^{2})^{1/3} dx$$  &  $$\displaystyle I_{2}=\int_{0}^{1}(1-x^{3})^{1/2} dx$$

On the basis of above information, answer the following questions: 

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer