Mathematics

Evaluate the following integral
$$\int { \cfrac { \cos { x }  }{ \cos { \left( x-a \right)  }  }  } dx\quad $$


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

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

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

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

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

$$I=x\cos{a}-\sin{a}\log{\left|\sec{\left(x-a\right)}\right|}+c$$
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Subjective Medium Published on 17th 09, 2020
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