Mathematics

# Solve $\displaystyle\int {\dfrac{{{{\cos }^2}\theta d\theta }}{{{{\cos }^2}\theta + 4{{\sin }^2}\theta }}}$.

##### SOLUTION
Solution :-
$\displaystyle I=\int \frac{cos^{2}x}{cos^{2}x+4sin^{2}x}dx$

$\displaystyle =\int \frac{cos^{2}x}{cos^{2}x+4sin^{2}x}.\frac{sec^{4}x}{sec^{4}x}dx$

$\displaystyle =\int \frac{sec^{2}x\:dx}{sec^{2}x+4tan^{2}xsec^{2}x}$

$\displaystyle =\int \frac{sec^{2}x\:dx}{sec^{2}x(1+4tan^{2}x)}$

Now $tan x=t$

$sec^{2}xdx=dt$

$\displaystyle \therefore I=\int \frac{dt}{(1+t^{2})(1+4t^{2})}=\int (\frac{A}{1+t^{2}}+\frac{B}{1+4t^{2}})dt$

Now comparing $1=A(1+4t^{2})+B(1+t^{2})$

$\Rightarrow A+B=1$   $4A+B=0$

$\therefore A= -1/3$    $B= 4/3$

$\displaystyle I= \frac{-1}{3}\int \frac{1}{1+t^{2}}dt+\frac{4}{3}\int \frac{1}{1+4t^{2}}dt$

We know, $\displaystyle \int \dfrac{1}{{a^2+x^2}}dx =\dfrac {1}{a}\tan^{-1}\left(\dfrac {x}{a}\right)+c$

$I\displaystyle =\frac{-1}{3}tan^{-1}t+\frac{4}{3}\frac{tan^{-1}(2t)}{2}+c$

Putting the value of $t$ as $\tan x$

$I\displaystyle =\frac{-1}{3}tan^{-1}(tanx)+\frac{2}{3}tan^{-1}(2tanx)+c$

$\displaystyle =\frac{-1}{3}x+\frac{2}{3}tan^{-1}(2tanx)+c$

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

#### Realted Questions

Q1 Single Correct Medium
If $\int _0^1 xdx = \dfrac {\pi}{4} - \dfrac {1}{2} ln 2$ then the value of definite integral $\int _0^1 \tan^{-1} (1-x+x^2) dx$ equals :
• A. $\dfrac {\pi}{4} + ln 2$
• B. $\dfrac {\pi}{4} - ln2$
• C. $2 ln 2$
• D. $ln2$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\int_{0}^{\frac{\pi}{2}} \sin \theta \cdot \sin 2 \theta d \theta$
• A. $\frac{4}{3}$
• B. $\frac{1}{3}$
• C. $\frac{3}{4}$
• D. $\frac{2}{3}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
The integral $\int \dfrac {2x^{12} + 5x^{9}}{(x^{5} + x^{3} + 1)^{3}} dx$ is equal to:
• A. $\dfrac {-x^{5}}{(x^{5} + x^{3} + 1)^{2}} + C$
• B. $\dfrac {x^{5}}{2(x^{5} + x^{3} + 1)^{2}} + C$
• C. $\dfrac {-x^{-10}}{2(x^{5} + x^{3} + 1)^{2}} + C$
• D. $\dfrac {x^{10}}{2(x^{5} + x^{3} + 1)^{2}} + C$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Solve $\displaystyle\int { \dfrac { { e }^{ 6\log { x } }-{ e }^{ 5\log { x } } }{ { e }^{ 4\log { x } }-{ e }^{ 3\log { x } } } } dx$

Let $\displaystyle f\left ( x \right )=\frac{\sin 2x \cdot \sin \left ( \dfrac{\pi }{2}\cos x \right )}{2x-\pi }$