Question

1/ cos ( x- a ) × cos ( x - b ) ​

Answer 1

Answer:

Step-by-step explanation:

Check that if you multiply and divide the integrand by [math]\sin (a-b)[/math], you would be able to simplify it with a bit of manipulation.

[math]\displaystyle {\frac 1{\sin (a-b)} \cdot \dfrac {\sin \left( (x-b)-(x-a) \right)}{\cos (x-a) \cos (x-b)}}[/math]

Note that the term in the numerator can be expanded to obtain

[math]\displaystyle {\sin (x-b) \cos (x-a) - \cos (x-b) \sin (x-a)}[/math]

Then by cancelling like terms in the numerator and the denominator, we are only left with

[math]\displaystyle {\frac 1{\sin (a-b)}\cdot\left( \tan (x-b) - \tan (x-a) \right)}[/math]

The integration after which is basic.

[math]\displaystyle {\boxed {\frac 1{\sin (a-b)} \cdot \ln \left| \frac {\cos (x-a)}{\cos (x-b)} \right| +C}}[/math]

Answer 2

It is one of those peculiar integrals in the CBSE curriculum which would require you to memorise a crucial trick in order to quickly get through in a few steps.

1cos(−)cos(−)Note that the term in the numerator can be expanded to obtain

sin(−)cos(−)−cos(−)sin(−)

Then by cancelling like terms in the numerator and the denominator, we are only left with

1sin(−)⋅(tan(−)−tan(−))

1sin⁡(a−b)⋅

(tan⁡(x−b)−tan(x-a)

The integration after which is basic.

1sin(−)⋅ln∣∣∣cos(−)cos(−)∣∣∣+

1sin⁡(a−b)⋅

ln⁡ cos⁡(x−a)cos⁡(x−b)|+C