Mathematics

Solve  $$\displaystyle \int { \dfrac { \left( { x }^{ 2 }+1 \right) \left( { x }^{ 2 }+2 \right)  }{ \left( { x }^{ 2 }+3 \right) \left( { x }^{ 2 }+4 \right)  }  } dx$$


SOLUTION
$$\int{\dfrac{(x^2+1)(x^2+2)}{(x^2+3)(x^2+4)}}dx$$

This can be written as 

$$\Rightarrow \int{(\dfrac{-4x^2-10}{(x^2+3)(x^2+4)}}+1)dx$$

$$\Rightarrow \int{\dfrac{-4x^2-10}{(x^2+3)(x^2+4)}}dx+\int 1dx$$

$$\Rightarrow -2\int{\dfrac{2x^2+5}{(x^2+3)(x^2+4)}}dx+\int 1dx$$

$$\Rightarrow -2\int{\dfrac{3}{x^2+4}}dx+2\int{\dfrac{1}{x^2+3}}dx+x$$

$$\Rightarrow -6\int{\dfrac{1}{x^2+4}}dx+2\int{\dfrac{1}{x^2+3}}dx+x$$

$$\Rightarrow -\dfrac{6}{2}\tan ^{-1}(\dfrac{x}{2})+\dfrac{2}{\sqrt 3}\tan ^{-1}(\dfrac{x}{\sqrt 3})+x$$

$$\Rightarrow x -3\tan ^{-1}(\dfrac{x}{2})+\dfrac{2}{\sqrt 3}\tan ^{-1}(\dfrac{x}{\sqrt 3})$$
View Full Answer

Its FREE, you're just one step away


Subjective Hard Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 114
Enroll Now For FREE

Realted Questions

Q1 Subjective Medium
Show that $$\displaystyle \int \frac{dx}{\sqrt{\left (1+e^{x}+e^{2x}  \right )}}=\log \left \{ \left ( z+\frac{1}{2} \right )+\sqrt{\left ( z^{2}+z+1 \right )} \right \}$$. where $$z=\displaystyle \frac{1}{e^{x}}$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Medium
$$\displaystyle\int \frac{x\sin ^{-1}x}{\sqrt{\left ( 1-x^{2} \right )}}dx.$$
  • A. $$\displaystyle \sqrt{1-x^{2}}\sin ^{-1}x+x.$$
  • B. $$\displaystyle -\sqrt{1-x^{2}}\cos ^{-1}x+x.$$
  • C. $$\displaystyle -\sqrt{1-x^{2}}\sin ^{-1}x-x.$$
  • D. $$\displaystyle -\sqrt{1-x^{2}}\sin ^{-1}x+x.$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Single Correct Hard
The value of $$\displaystyle \int_{ {\pi^{3}}/{27}}^{ {\pi^{3}}/{8}}sinx.dt$$ , where $$t= x^3$$, is?
  • A. $$cos\displaystyle \frac{\pi^{3}}{27}-cos\displaystyle \frac{\pi^{3}}{8}$$
  • B. $$\displaystyle \frac{\pi^2}{6}$$
  • C. None of these
  • D. $$\displaystyle \frac{\pi^{2}}{6}+(3-\sqrt3)\pi-3$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Integrate with respect to x:
(i) $$\dfrac{x}{nx}$$
(ii) $$x\sin^2x$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Medium
Consider two differentiable functions $$f(x), g(x)$$ satisfying $$\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$$ & $$\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$$. where $$\displaystyle f(x)>0    \forall  x \in  R$$

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

Asked in: Mathematics - Limits and Derivatives


1 Verified Answer | Published on 17th 08, 2020

View Answer