Mathematics

Find: $$\displaystyle \int{\dfrac{4}{(x-2)(x^{2}+4)}dx}$$


SOLUTION
$$\begin{array}{l} Given, \\ \int { \frac { 4 }{ { \left( { x-2 } \right) \left( { { x^{ 2 } }+4 } \right)  } } dx }  \\ Now, \\ 4.\int { \frac { 1 }{ { \left( { x-2 } \right) \left( { { x^{ 2 } }+4 } \right)  } } dx }  \\ Apply\, \, x=2u \\ 4.\int { \frac { 1 }{ { 4\left( { u-2 } \right) \left( { { u^{ 2 } }+4 } \right)  } } dx }  \\ \frac { 1 }{ { 4\left( { u-2 } \right) \left( { { u^{ 2 } }+4 } \right)  } } :\frac { { -u-1 } }{ { 2\left( { { u^{ 2 } }+1 } \right)  } } +\frac { 1 }{ { 2\left( { u-1 } \right)  } }  \\ =4.\frac { 1 }{ 4 } \left( { \int { \frac { { -u-1 } }{ { 2\left( { { u^{ 2 } }+1 } \right)  } } du+\frac { 1 }{ { 2\left( { u-1 } \right)  } } du }  } \right)  \\ \int { \frac { { -u-1 } }{ { 2\left( { { u^{ 2 } }+1 } \right)  } } du=\frac { 1 }{ 2 } \left( { -\frac { 1 }{ 2 } In\left| { { u^{ 2 } }+1 } \right|  } \right)  }  \\ \int { \frac { 1 }{ { 2\left( { { u^{ 2 } }+1 } \right)  } } du } =\frac { 1 }{ 2 } In\left| { { u^{ 2 } }+1 } \right|  \\ =4.\frac { 1 }{ 4 } \left( { \frac { 1 }{ 2 } \left( { -\frac { 1 }{ 2 } In\left| { { u^{ 2 } }+1 } \right|  } \right) -\tan  \left( u \right) +\frac { 1 }{ 2 } \left| { { u^{ 2 } }+1 } \right|  } \right)  \\ =\frac { 1 }{ 2 } \left( { -\frac { 1 }{ 2 } In\left| { { { \frac { x }{ 4 }  }^{ 2 } }+1 } \right| -\tan  \left( { \frac { x }{ 2 }  } \right) +\frac { 1 }{ 2 } \left| { \frac { x }{ 2 } -1 } \right|  } \right)  \\ =\frac { 1 }{ 2 } \left( { -\frac { 1 }{ 2 } In\left| { { { \frac { x }{ 4 }  }^{ 2 } }+1 } \right| -\tan  \left( { \frac { x }{ 2 }  } \right) +\frac { 1 }{ 2 } \left| { \frac { x }{ 2 } -1 } \right|  } \right) +C \end{array}$$
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 Single Correct Medium
For $$n\in N$$, $$0 < t < \pi/2$$; the value of $$\displaystyle\int^{n\pi +t}_0(|\cos x|+|\sin x|)dx=$$?
  • A. $$4n+\sin t-\cos t +1$$
  • B. $$4n-\sin t-\cos t+1$$
  • C. $$2n-\sin t-\cos t+1$$
  • D. $$2n-\sin t-\cos t-1$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Hard
Solve $$\int_0^1 {{{\cot }^{ - 1}}} (1 - x + {x^2})dx$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Evaluate : $$\displaystyle \int \dfrac { \cos x } { ( 1 + \sin x ) ^ { 2 } ( 2 + \sin x ) } d x$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Evaluate: $$\displaystyle \int {{\dfrac{\cos 2x - \cos 2\alpha }{\cos x - \cos \alpha }}} dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Easy
Evaluate:
$$ \int_{}^{} {\frac{{ - 1}}{{\sqrt {1 - {x^2}} }}dx} $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer