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Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals3 Marks
Question
Find the integral of the function ${\frac{{\cos 2x - \cos 2\alpha }}{{\cos x - \cos \alpha }}}$
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Answer

$\int {\frac{{\cos 2x - \cos 2\alpha }}{{\cos x - \cos \alpha }}} dx$

$= \int {\frac{{\left( {2{{\cos }^2}x - 1} \right) - \left( {2{{\cos }^2}\alpha - 1} \right)}}{{\left( {\cos x - \cos \alpha } \right)}}} dx$

$= \int {\frac{{\left( {2{{\cos }^2}x} \right) - \left( {2{{\cos }^2}\alpha} \right)}}{{\left( {\cos x - \cos \alpha } \right)}}} dx$

$ = \int {\frac{{2\left( {\cos^2 x - \cos^2 \alpha } \right)}}{{\left( {\cos x - \cos \alpha } \right)}}} dx$

$ = \int {\frac{{2\left( {\cos x + \cos \alpha } \right)\left( {\cos x - \cos \alpha } \right)}}{{\left( {\cos x - \cos \alpha } \right)}}} dx$
$ = \int 2({cos x + cos \alpha })\ dx$
$= 2\left( {\sin x + x\cos \alpha } \right) + c$.

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