Mathematics

# Find: $\int { \frac { cos\theta }{ (4+sin^{ 2 }\theta )(5-4{ cos }^{ 2 }\theta ) } d\theta }$

##### SOLUTION
$\displaystyle\int \dfrac{\cos \theta}{(4+\sin ^2 \theta)(5-4\cos^2 \theta)}d \theta=\int \dfrac{\cos \theta}{(4+\sin ^2 \theta)(5-4+4\sin ^2 \theta)}d\theta=\int \dfrac{\cos \theta}{(4+\sin^2 \theta)(1+4\sin ^2 \theta)}d\theta$
Put $t=\sin \theta\implies d t=\cos \theta d\theta$
$\implies \displaystyle\int \dfrac{d t}{(t^2+4)(4 t^2+1)}=\int \dfrac{A d t}{t^2+4}+\int \dfrac{B dt}{4 t^2+1}$
$\implies A(4 t^2+1)+B(t^2+4)=1\implies 4 A+B=0,A+4 B=1$
$\implies A=-\dfrac{1}{15},B=\dfrac{4}{15}$
$\implies -\dfrac{1}{15}\displaystyle\int \dfrac{d t}{t^2+4}+\dfrac{1}{15}\int \dfrac{d t}{t^2+1/4}=-\dfrac{1}{30}\text{tan}^{-1}\bigg(\dfrac{t}{2}\bigg)+\dfrac{2}{15}\text{tan}^{-1}(2 t)+C=-\dfrac{1}{30}\text{tan}^{-1}\bigg(\dfrac{\sin \theta}{2}\bigg)+\dfrac{2}{15}\text{tan}^{-1}(2\sin \theta)+C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Subjective Medium
Evaluate the given integral.
$\displaystyle\int{\dfrac{{\left(1+x\right)}^{2}}{\sqrt{x}}dx}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int^{\pi/2}_{0}In (\sin x)dx$ equals
• A. $-\pi\ln 2$
• B. $-\dfrac{\pi }{4}\ln 2$
• C. $-\dfrac{\pi }{8}\ln 2$
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1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
If $\displaystyle I=\frac{\cot x}{\sqrt{a+b\cot^{2}x}}dx\left ( 0< a < b \right ),$ then I equals

• A. $\displaystyle \sqrt{b-a} \sin ^{-1}\left ( \sqrt{b-a}x \right )+Const$
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• C. $-{\sqrt{b-a}\cos^{-1}}\left ( \displaystyle \sqrt{\frac{b-a}{b}\sin x} \right )+Const$
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1 Verified Answer | Published on 17th 09, 2020

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The value of  $\int { \sqrt { \dfrac { { e }^{ x }-1 }{ { e }^{ x }+1 } } } dx$ is equal to

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
The value of $\int { { e }^{ \tan { \theta } } } \left( \sec { \theta } -\sin { \theta } \right) d\theta$ is equal to ?
• A. $-{ e }^{ \tan { \theta } }\sin { \theta } +C$
• B. ${ e }^{ \tan { \theta } }\sin { \theta } +C$
• C. ${ e }^{ \tan { \theta } }\sec { \theta } +C$
• D. ${ e }^{ \tan { \theta } }\cos { \theta } +C$