Mathematics

# $\displaystyle \int \dfrac{1}{3\sin x +4\cos x} dx$ equal, where $\tan \alpha = \dfrac{4}{3}$

$\dfrac{1}{5} ln \left|\tan \left(\dfrac{x}{2} + \dfrac{\alpha}{2}\right)\right|+K$

##### SOLUTION
$\displaystyle \int \dfrac{1}{3\sin x +4\cos x} dx = \int 5\dfrac{1}{\left(\dfrac{3}{5}\sin x + \dfrac{4}{5} \cos x\right)}dx$
$=\displaystyle \dfrac{1}{5} \int \dfrac{1}{\sin x\cdot \cos \alpha + \cos x \cdot \sin \alpha}dx$    $\tan \alpha = \dfrac{4}{3}$
$=\displaystyle \int \frac{1}{\sin (x+\alpha)}dx$                            $\sin \alpha = \dfrac{4}{5}$
$=\displaystyle \frac{1}{5} \int cose (x+\alpha) dx$                      $\cos \alpha = \dfrac{3}{5}$
$=\displaystyle \frac{1}{5} ln \left|\tan \left(\frac{\pi}{2} + \frac{d}{2}\right)\right| + k$

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

#### Realted Questions

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$\int \frac{xdx}{\sqrt{1+x^{2}+\sqrt{(1+x^{2})^{3}}}}$
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• B. $2(1+\sqrt{1+x^{2}})+C$
• C. $2\sqrt{1+\sqrt{1+x^{2}}}+C$
• D. $\frac{1}{2}ln(1+\sqrt{1+x^{2}})+C$

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