Mathematics

# Solve $\displaystyle\int {\dfrac{{dx}}{{2\sin x + \cos x + 3}}}$

$\tan^{-1}\left(\tan \dfrac{x}{2}+1\right)+c$

##### SOLUTION
$\displaystyle I=\int \dfrac{dx}{2\sin x+\cos x+3}$

$\displaystyle =\int \dfrac{dx}{2\left(\dfrac{2\tan\dfrac{x}{2}}{1+\tan^2\dfrac{x}{2}}\right)+\left(\dfrac{1-\tan^2\dfrac{x}{2}}{1+\tan^2\dfrac{x}{2}}\right)+3}$

$=\displaystyle \int \dfrac{1+\tan^2\dfrac{x}{2}dx}{4\tan\dfrac{x}{2}+1-\tan^2\dfrac{x}{2}+3+3\tan^2\dfrac{x}{2}}$

$=\displaystyle \int \dfrac{\sec^2\dfrac{x}{2}dx}{2\tan^2\dfrac{x}{2}+4\tan\dfrac{x}{2}+4}$

Let $\tan\dfrac{x}{2}=t$

$=\sec^2\dfrac{x}{2}.\dfrac{1}{2}dx=dt$

$=\boxed{\sec^2\dfrac{x}{2}dx=2dt}$

$=\displaystyle \int \dfrac{2dt}{2t^2+4t+4}=\int \dfrac{dt}{t^2+2t+2}$

$=\displaystyle \int\dfrac{dt}{(t+1)^2+(1)^2}=\tan^{-1}(t+1)+c$

$=\tan^{-1}\left(\tan \dfrac{x}{2}+1\right)+c$

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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
Integrate $\displaystyle \int {{e^{\sin x}}.\cos x\,\,dx}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
$\displaystyle \int\, sin^2\, (lnx)\, dx$ is equal to
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• C. $\displaystyle \frac{x}{10} (5 -2sin(2lnx) + sin(2lnx)) + c$
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1 Verified Answer | Published on 17th 09, 2020

Q3 One Word Hard
$\displaystyle \int \frac{\sin x}{\sin 4x}dx=-\frac{1}{k}\log \frac{1+\sin x}{1-\sin x}+\frac{1}{4\sqrt{2}}\log \frac{1+\sqrt{2}\sin x}{1-\sqrt{2}\sin x}$. Find the value of $k$.

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate the following integral:

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