Mathematics

Solve $$\int { \dfrac { 1 }{ \sqrt { { a }^{ 2 }+{ \left( x-c \right)  }^{ 2 } }  }  } $$


SOLUTION
$$ put \ x=c+atan\Theta $$
$$ {dx}=asec^{2}\Theta{d\Theta} $$
$$ Now  \ the \  inegration \ reduces \ to: $$
$$ \int{\dfrac {asec^{2}\Theta{d\Theta}}{asec\Theta}}  $$
$$  =\int{sec\Theta{d\Theta}}$$
$$ =ln(sec\Theta+tan\Theta) +C\to(1) $$
$$tan\Theta=\dfrac{x-c}{a}  \ and \ sec\Theta=\dfrac{\sqrt{a^{2}+(x-c)^{2}}}{a}  $$
$$ putting \ the \ values \ of \ sec\Theta \ and \ tan\Theta \ in (1)$$
$$ required \ answer \ will \  be \  ln(\dfrac{x-c}{a}+\dfrac{\sqrt{a^{2}+(x-c)^{2}}}{a})+C $$
$$  or; $$
$$ ln((x-c)+\sqrt{a^{2}+(x-c)^{2}})+C$$
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
Evaluate :
$$\int _{ 0 }^{ 4 }{ \left( 16-{ x }^{ 2 } \right) ^{ 1/2 } } dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Hard
Evaluate the integral
$$\displaystyle \int_{0}^{\pi}\frac { x\quad dx }{ 1+\cos ^{ 2 }{ x }  }$$
  • A. $$0$$
  • B. $$\displaystyle \frac{\pi^{2}}{4}$$
  • C. $$\displaystyle \frac{\pi^{2}}{8}$$
  • D. $$\displaystyle \frac{\pi^{2}}{2\sqrt{2}}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Integrate the function    $$\cfrac {5x+3}{\sqrt {x^2+4x+10}}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Evaluate the following integral:
$$\displaystyle \int_{0}^{3}(2x^2+3x +5)dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Medium
Prove that: $$\displaystyle\int^1_{-1}(ax^3+bx)dx=0$$ .

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer