Mathematics

Evaluate the following integral
$$\int { \cfrac { \cos { 2x }  }{ { \left( \cos { x } +\sin { x }  \right)  }^{ 2 } }  } dx\quad $$


SOLUTION
$$I=\displaystyle\int{\dfrac{\cos{2x}}{{\left(\cos{x}+\sin{x}\right)}^{2}}dx}$$

$$I=\displaystyle\int{\dfrac{{\cos}^{2}{x}-{\sin}^{2}{x}}{{\left(\cos{x}+\sin{x}\right)}^{2}}dx}$$

$$I=\displaystyle\int{\dfrac{\left(\cos{x}+\sin{x}\right)\left(\cos{x}-\sin{x}\right)}{{\left(\cos{x}+\sin{x}\right)}^{2}}dx}$$

$$I=\displaystyle\int{\dfrac{\left(\cos{x}-\sin{x}\right)}{\left(\cos{x}+\sin{x}\right)}dx}$$

Let $$t=\cos{x}+\sin{x}\Rightarrow\,dt=\left(-\sin{x}+\cos{x}\right)dx$$

$$I=\displaystyle\int{\dfrac{dt}{t}}$$

$$I=\log{\left|t\right|}+c$$

$$\therefore\,I=\log{\left|\cos{x}+\sin{x}\right|}+c$$ where $$t=\cos{x}+\sin{x}$$
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Subjective Medium Published on 17th 09, 2020
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