Mathematics

# The integral $\displaystyle\int \dfrac{\sin^2x \cos^2x}{(\sin^5x+ \cos^3x \sin^2 x+ \sin^3x \cos^2x + \cos^5x)^2}dx$

$\dfrac{-1}{3(1+ \tan^3x)}+c$

##### SOLUTION
we have to evaluate,
$\Rightarrow I=\int\dfrac{sin^2x\ cos^2\ x}{(sin^5x+cos^3x\sin^2x+sin^3x\ cos^2x+cos^5x)^2}dx$

$\Rightarrow \int\dfrac{sin^2 x\ cos^2x}{\begin{Bmatrix}(sin^2x(sin^3x+ cos^3x)+cos^2x(sin^3x+ cos^3x)\end{Bmatrix}^2}dx$

$\Rightarrow \int\dfrac{sin^2 x\ cos^2x}{\begin{Bmatrix}(sin^2x+ cos^2x)(sin^3x+ cos^3x)\end{Bmatrix}^2}dx$

$\Rightarrow \int\dfrac{sin^2 x\ cos^2x}{\begin{Bmatrix}(sin^3x+ cos^3x)\end{Bmatrix}^2}dx$

$\Rightarrow \int\dfrac{sin^2 x\ cos^2x}{\begin{Bmatrix}sin^6x+2sin^3xcos^3x+ cos^6x\end{Bmatrix}}dx$

on dividing numerator and denominator by $cos^6x$we get

$\Rightarrow \int\dfrac{tan^2 x\ sec^2x}{\begin{Bmatrix}tan^6x+2tan^3x+1\end{Bmatrix}}dx$

$\Rightarrow \int\dfrac{tan^2 x\ sec^2x}{\begin{pmatrix}1+tan^3x\end{pmatrix}^2}dx$

Let $(1+ tan^3x )=t\Rightarrow 3tan^2x\ sec^2xdx =dt$

$\Rightarrow \dfrac13\int\dfrac{dt}{\begin{pmatrix}t\end{pmatrix}^2}= -\dfrac {1}{3}\ \dfrac1t + C$

$=\dfrac{-1}{3(1+ tan^3x)}+ C$

$\therefore \text{option A is correct}$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Subjective Medium
Prove that$\displaystyle\int \frac{dx}{\sqrt{\left [ \left ( x-a \right ) \right ]\left ( x-b \right )}}$ equal to $\log { \left( -2\left( \sqrt { \left( a-x \right) \left( b-x \right) } +x \right) +a+b \right) +c }$.

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Write the anti derivative of
$\left( 3\sqrt { x } +\cfrac { 1 }{ \sqrt { x } } \right)$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Find: $\displaystyle\int { \dfrac { { \left( { x }^{ 4 }-x \right) }^{ \dfrac { 1 }{ 4 } } }{ { x }^{ 5 } } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Solve:-
$\displaystyle\int {\dfrac{{{e^x}(1 + x)}}{{{{\cos }^2}(x{e^x})}}} dx$
• A. $I=2\tan \left( x{{e}^{x}} \right)+C$
• B. $I=\tan \left( {{e}^{x}} \right)+C$
• C. None of these
• D. $I=\tan \left( x{{e}^{x}} \right)+C$

Let $F: R\rightarrow R$ be a thrice differential function. Suppose that $F(1) = 0, F(3) = -4$ and $F'(x)<0$ for all $x\in\left(\dfrac{1}{2},3\right)$. Let $f(x) = xF(x)$ for all $x\in R$.