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 $$t=\sin \theta\implies d t=\cos \theta d\theta$$
$$\implies \displaystyle\int \dfrac{d t}{(t^2+4)(4 t^2+1)}=\int \dfrac{A d t}{t^2+4}+\int \dfrac{B dt}{4 t^2+1}$$
$$\implies A(4 t^2+1)+B(t^2+4)=1\implies 4 A+B=0,A+4 B=1$$
$$\implies A=-\dfrac{1}{15},B=\dfrac{4}{15}$$
$$\implies -\dfrac{1}{15}\displaystyle\int \dfrac{d t}{t^2+4}+\dfrac{1}{15}\int \dfrac{d t}{t^2+1/4}=-\dfrac{1}{30}\text{tan}^{-1}\bigg(\dfrac{t}{2}\bigg)+\dfrac{2}{15}\text{tan}^{-1}(2 t)+C=-\dfrac{1}{30}\text{tan}^{-1}\bigg(\dfrac{\sin \theta}{2}\bigg)+\dfrac{2}{15}\text{tan}^{-1}(2\sin \theta)+C$$
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 Subjective Medium
Evaluate the given integral.
$$\displaystyle\int{\dfrac{{\left(1+x\right)}^{2}}{\sqrt{x}}dx}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
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. $$ -\dfrac{\pi }{8}\ln 2 $$
  • D. $$ -\dfrac{\pi }{2}\ln 2 $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
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$$
  • B. $$\displaystyle \sqrt{b-a}\sin ^{-1}\left ( \sqrt{b-a}\sin x \right )+Const$$
  • C. $$-{\sqrt{b-a}\cos^{-1}}\left ( \displaystyle \sqrt{\frac{b-a}{b}\sin x} \right )+Const$$
  • D. $$\displaystyle \frac{1}{\sqrt{b-a}}\sin ^{-1}\left ( \sqrt{\frac{b-a}{b}\sin x} \right )+Const$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
The value of  $$\int { \sqrt { \dfrac { { e }^{ x }-1 }{ { e }^{ x }+1 }  }  } dx$$ is equal to

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
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. $${ e }^{ \tan { \theta } }\cos { \theta } +C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer