1/(x-a)(x-b) - NCERT Class XII Mathematics Chapter -

NCERT Class XII Mathematics Chapter - - Solutions

Exercise - 7.3
Find the integrals of the functions in Exercises 62 to 83.

Question : 83 of 271

Marks: +1, -0

\frac{1}{\cos(x-a)\cos(x-b)}

Solution:

Let

I=\int \frac{1}{\cos(x-a)\cos(x-b)}dx

=\frac{1}{\sin(a-b)}\int \frac{\sin[(x-b)-(x-a)]}{\cos(x-a)\cos(x-b)}dx

=\frac{1}{\sin(a-b)}\times\int\left(\frac{\sin(x-b)\cos(x-a)-\cos(x-b)\sin(x-a)}{\cos(x-a)\cos(x-b)}\right)dx\qquad\left[\because \sin(A-B)=\sin A\cos B-\cos A\sin B\right]

=\frac{1}{\sin(a-b)}\left[\int \tan(x-b)dx-\int \tan(x-a)dx\right]

=\frac{1}{\sin(a-b)}\left[-\log\left|\cos(x-b)\right|+\log\left|\cos(x-a)\right|\right]+C

=\frac{1}{\sin(a-b)}+\log\left|\frac{\cos(x-a)}{\cos(x-b)}\right|+C

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